This client already tracked the net profit and loss generated by every customer who used his services, and had begun to analyze his customers by group. He was especially interested in Customer Type A; his gut instinct told him that Type A customers were quite profitable compared to the others (Type B) and he wanted to back up this feeling with numbers.

He found that, on average, Type A customers generate about $92 profit per month, and Type B customers average about $115 per month (The data and figures that we are using in this discussion aren’t actual client data, of course, but a notional example). He also found that while Type A customers make up about 4% of the customer base, they generate less than 4% of the net profit per month. So Type A customers actually seem to be less profitable than Type B customers. Apparently, our client was mistaken.

Or was he?

A little more elementary statistics revealed that the median profit generated by Type A customers is $65 — e.g., half the customers from group A generate more than $65 profit per month. The median for Type B customers is about $4.80 — so half the customers from group B generate less than five dollars profit every month. Maybe our client’s instincts aren’t completely off-base.

Let’s look at the distribution of net profit across both customer populations:

Figure 1: Distribution of net profit for Type A customers (blue) and Type B customers (red). The x-axis gives the net profit or loss, and the y-axis gives the fraction of the population that generates a given net profit.

This pattern is typical among the customers of many businesses. The majority of customers generate relatively moderate profit (or loss); but there is an important minority of large-profit and large-loss customers out on both tails. In this case, the monthly customer value actually ranges from losses in the tens of thousands to profits of several hundred thousands (I clipped the graph, for “clarity”).

I hesitate to call these large magnitude customers “outliers” because that term implies anomalous, possibly erroneous, data. In this case, the “outliers” are relatively rare, but important, customers who can potentially make the difference between a company that is in the black or in the red. Still, they are the exception and their behavior doesn’t necessarily tell you anything about the behavior of your typical customer. Knowing the mean profitability of a given customer group is important, of course, but the estimate will be dominated by your exceptionally profitable or lossy customers in that group, and as we’ve seen, that hides information about the majority of your customers.

You might remember from our Good Graphs article that if you have positive skewed data with a wide dynamic range, graphing the data on a log scale helps you see phenomena across the entire range of data that you might miss on the ordinary graph. Unfortunately, we have data here in the positive and negative range. So let’s split the customers into three groups: profitable, unprofitable, and break-even. About 5-6% of the customer base is break-even, roughly the same proportion in Groups A and B; we’ll ignore them for now, and look at the profitable customers first (over 80% of the customers, in both groups).

Figure 2: Distribution of profit from profitable Type A customers (blue) and Type B customers (red). The x-axis gives net profit on a log 10 scale, so every labelled tick corresponds to a change by a factor of 100 (eg. 10^0 = $1, 10^2 = $100, and so on). The y-axis represents the fraction of the profitable customer base that generates a given profit.

Now we can clearly see that (among profitable customers) the typical Type A customer is in fact more profitable than the typical Type B customer. The mean profit from profitable Type A customers is about $227, and the median profit is about $93 (shown by the dashed blue line). About 2/3 of the profitable Type A customers generate between $21 and $400 in profit, and over 95% of them generate between $5 and $1721 in profit. We can call that 95% the set of “typical” profitable Type A customers. That’s not a standard definition, but it’s an intuitive one, and useful for this discussion.

Approximately 2.5% of Type A customers generate profits greater than $1721; let’s call them the Type A “best-customers,” some of whom generate profits in the tens of thousands. They are responsible for 30% of the profit that comes from profitable Type A customers, and 3% of the profit that comes from all profitable customers (even though they only make up 0.2% of that population).

Profitable Type B customers generate $148 mean profit, and about $7.67 median profit (the red dashed line). A typical profitable Type B customer generates between six cents and $1031 in profit — a lower range than what the typical Type A customer generates, although the very highest-performing Type B customers are competitive with the highest-performing Type A customers (about 130 Type B customers outperform all the Type A customers).

Unfortunately, when Type A customers are unprofitable, they are typically more unprofitable than those of Type B. This is another reason why the mean profit from Type A customers overall was so low. Our client correctly perceived that Type A customers are typically quite profitable, but there is a small population of real clunkers in the group, too.

Figure 3: Distribution of loss from unprofitable Type A customers (blue) and Type B customers (red). The x-axis gives loss on a log 10 scale; further to the right on the graph means a larger loss. An unprofitable Type A customer loses a median of $137 a month, and a mean of $1180. Unprofitable Type B customers lose a median of $4.80, and a mean of $210.

We can do a similar analysis for the entire base of profitable customers. We would find that the typical profitable customer generates between six cents and $1200 in profit every month (median $8.65, mean $153), and that the 2.5% of best-customers generate over 60% of the profits.

The Lognormal Distribution

Figure 4: (Left) Distribution of profitable customers (graph clipped at $10,000). The x-axis gives the net profit, and the y-axis gives the fraction of the population that generates a given net profit. (Right) Distribution of profitable customers plotted on a log scale.

The distribution of highly skewed positive data, like the value of profitable customers, incomes, sales, or stock prices, can often be modelled as a lognormal distribution: that is, the log of the data is distributed in a bell-shaped curve centered (in log space) at the median of the data (remember, for a normal curve, the median and the mean are the same). In our case, both the profits (seen above, in Figure 4) and the losses are distributed approximately lognormally. For lognormal populations, the mean is generally much higher than the median, and the bulk of the contribution towards the mean will be made by a small population of highest-valued data points. If you use the mean as a stand-in for value, you will overstate the value of most of your customers.

If your customer value data is distributed approximately lognormally, then you can quickly estimate the range of values that 95% of your customers will fall into. About 95% of normally distributed data will fall within plus/minus two standard deviations of the mean, and taking logarithms converts multiplication into addition. So: if sd is the standard deviation of the natural log of your customer value data, M is the median profit, and k = exp(sd), then 95% of your customers will fall in the value range (M/(k*k), M*k*k). The 2.5% of customers who generate more than M*k*k profit are your best-customers, who often drive a majority of your profit.

Long Tail Theory

The distribution of customers above sounds a lot like Chris Anderson’s Long Tail Theory of consumer goods. Most of the revenue of (for example) a bookseller or a music store comes from a few “hits”, or blockbusters, with the rest of the merchant’s inventory out along the tail of Figure 5, moving a relatively small volume per title.

Figure 5: (Top) A notional long tail curve. The y-axis represents sales volume, and the x-axis represents goods ranked from most to least popular. The highest selling goods are to the left. Note that this figure represents the sales curve differently from how the distribution of customer value is represented on the left side of Figure 4. (Bottom) The customer value data (top 10,000 customers) from Figure 4, plotted in the style above. The y-axis has been limited to $50,000 for clarity.

Anderson generally assumes that sales of such goods are distributed as a power law distribution, rather than a lognormal; the log of power law data isn’t distributed symmetrically, but actually has a longer tail to the right. This means that even for the log of the data, the mean is higher than the median. In fact, in some cases, the mean of a power law distribution can be infinite. If sales volume is power law distributed, then top-selling hits are responsible for an even larger percentage of total sales volume than would be the case with a lognormal.

The Pareto Distribution, which is one form of a power distribution, has been proposed as an alternative to the lognormal for modelling income distribution and other similar phenomena. Researchers have debated whether lognormal or Pareto is a better model for income distribution since at least the 1950s. Qualitatively, the two distributions have similar behavior. There are certain estimation and forecasting tasks where it does make a difference if your data follows a power law rather than a lognormal, but for the purposes of this discussion, it doesn’t really matter. For those who are interested, Michael Mitzenmacher has a fairly approachable discussion about the difference between power laws and lognormal distributions.

Back to Long Tail Theory. Historically, merchants tend to concentrate on high-volume items, due to space limitations and the cost of holding inventory. Overall, however, the sum total of tail-product sales will add up to a respectable volume, especially for web retailers who have unlimited “floor space” — or so the Long Tail theory goes. A retailer must then decide whether to follow the traditional “hits-oriented” strategy, or a more “tail-oriented” strategy that caters to the numerous niche markets.

If we draw an analogy with customer value, then best-customers are “hits.” Obviously, our client would like to “fire” his unprofitable customers while retaining his best-performing customers, and even attract more customers like them. But what about his little customers — the 95% of customers in the typical range? If his retention and growth strategy focuses primarily on attracting and retaining big customers, he is following a hits-oriented strategy. If his campaign also includes reaching out to little customers, then he is following something analogous to a tail strategy.

Not all business works like a music or book seller; the appropriate strategy will vary. Still, we can think of a few reasons why keeping little customers happy is a good idea.

For one thing, big customers are not only rare, but they are the ones that your competitors covet the most. Little customers, meanwhile, can still add up to a respectable chunk of change (close to 40% of net profit in our example above). A solid cushion of smaller customers may soften the blow to your profit margin, should a few of your bigger customers defect.

Consider computer sales. Microsoft and Dell serve both the corporate and consumer markets. To judge from their past marketing practices, they consider business customers to be the more valuable segment (see here for a rant somewhat related to this topic). But business IT sales have declined in the current moribund economic climate; analysts attribute the growth in computer sales for the last quarter of 2009 primarily to consumer spending. Dell’s market growth for that last quarter was much lower than that of HP, Acer, and Apple, which are more consumer-oriented companies. It’s also worth noting that Microsoft saw a 14% decline in revenue for the quarter ending September 30, 2009, compared to the year-ago quarter (and their earnings were in large part due to sales of the Xbox, a consumer product), while at the same time, consumer-oriented Apple saw a 24% increase in revenue from its year-ago quarter.

Your pool of little customers is also a pool of potential future best-customers. And you can’t always guess which ones. So a wise strategy might be to allocate part of your retention and growth campaign to providing loyalty incentives to smaller customers, and educating them about how your higher-end services or products might benefit them. Those little customers who have the means or opportunity to move on to the next level might very well appreciate your efforts, and stay with you, rather than defecting to a competitor.

Optimizing Sales vs. Optimizing Customers

One last thought about retail hits and high-value customers. McPhee’s Theory of Exposure, which is cited by Anita Elberse in her Harvard Business Review article “Should You Invest in the Long Tail?”, states that the popularity of music, film, TV or books is largely driven by “marginal audience participants” — the casual, or light, consumer. Casual consumers gravitate to already popular products because they have limited exposure to alternatives, and hence limited knowledge of them. Consumers of more obscure products, on the other hand, tend to be heavy (and knowledgable) consumers: voracious readers, dedicated music or film buffs, or enthusiasts of specific genres, like science-fiction or horror.

McPhee’s research was done in 1963, using subjects who had a fairly small range of choices, compared to internet scale. Elberse found, however, that the phenomena McPhee described still held for the internet merchants that she studied. She uses this observation (along with McPhee’s companion theory of Double Jeopardy) to argue that retailers should not substantially alter their traditional hits-based strategies. There is an alternative interpretation:

If your business follows McPhee’s theory, then hit products disproportionately attract low-value (low-volume) customers, and vice-versa.

So an overly hits-oriented strategy will skew you towards a base of low-value customers. Indeed, Seth Godin argues that iTunes and Amazon, who are in a better position to implement a more tail-oriented strategy, are thriving at the expense of physical stores exactly because they have been able to steal the quality (high-volume) customers away.

The moral is that both sales and customer value live in a lognormal world, where blockbuster products are marketed to a large cloud of low revenue customers, and high revenue best-customers are supported by large catalogues of low volume products. Fail to serve one side of this relationship, and you risk losing the other side.

Related posts:

1. Statistics to English Translation, Part 2b: Calculating Significance
2. A Demonstration of Data Mining
3. Good Graphs: Graphical Perception and Data Visualization

As in the last article, we will concentrate on situations where we want to test the difference of means. You should read that previous article first, so you are familiar with the terminology that we use in this one.

A pdf version of this current article can be found here.

How is Significance Determined?

Generally speaking, we calculate significance by computing a test statistic from the data. If we assume a specific null hypothesis, then we know that this test statistic will be distributed in a certain way. We can then compute how likely it is to observe our value of the test statistic, if we assume that the null hypothesis is true.

We’ll explain the use of a test statistic with our Sneetch example from the last installment.

The t-test for Difference of Means

Suppose that the test scores for both Star-Bellies and Plain-Bellies are normally distributed, with the means and standard deviations as given in the table below.

	(number of subjects)	(mean score)	(standard error)
Star-Bellies	50	78	7
Plain-Bellies	40	74	8

Remember from the previous installment that we can estimate the true population means and as normally distributed around the empirical population means and respectively, with variances
and
. This is shown in Figure 1. Informally speaking, there is no significant difference in the two populations if the shaded overlap area in Figure 1 is large.

Figure 1: The estimates of the means for two populations

Calculating this area is somewhat involved. Instead, we calculate the t-statistic:

(1)

where is called the pooled variance of the two populations.

(2)

For our Sneetch example, , and , or the negative of that, depending on which group is Group 1. There are
degrees of freedom.

If the null hypothesis is true, and the two populations are identical, then is distributed according to Student’s distribution with
degrees of freedom. Student’s distribution is sort of a “stretched out” bell curve; as the degrees of freedom increase (
), Student’s distribution approaches the standard normal distribution, ¹.

In other words, if the null hypothesis is true, should be near zero. The probability of seeing a of a certain magnitude or greater under the null hypothesis is given by the area under the tails of Student’s distribution:

Figure 2: The area under the tails for a given

This area is . For the Sneetch example, .

The further out on the tails is, the stronger the evidence that you should reject the null hypothesis. If you know for some reason that the mean of one population will be greater than or equal to the other, than you can use the one-tailed test:

Figure 3: The one-tailed test for a given

This test halves the p-value as compared to the two-tailed test, making a given value twice as significant. When in doubt about which to use, the two-tailed test is more conservative against false positives².

In discussions of t-tests, you will often see statements of the form:

The t-test meets the hypothesis that two means are equal if

for a two-tailed test, or

for a (right-sided) one-tailed test.

The quantities on the right hand side of the two equations above are called the critical values for a given significance level (usually,
) and degrees of freedom. The critical values are the values for which the area of the right hand tail is equal to .

Figure 4: Critical value for a one-tailed test. Reject the null hypothesis if

For a two-tailed test, you must halve the area under a single tail.

Figure 5: Critical value for a two-tailed test. Reject the null hypothesis if

This convention dates back to the time when computational resources were scarce, and researchers had to use pre-computed tables of critical values, rather than calculating directly. Today, general statistical packages such as R or Matlab can compute the CDFs of any number of standard distributions; once you can compute the CDF, directly computing (the area under the tails) is straightforward. Despite this, many tutorials of the t-test (and of the F-test, and other significance tests) still adhere to the convention of comparing test statistics to critical values. This tends to needlessly ritualize the whole process, and make it seem more complicated and mysterious than it actually is, at least in my opinion.

David Freedman was very much against the continued practice of using critical values, rather than reporting the actual p-value. The last chapter of Freedman, Pisani and Purves [FPP07] is worth reading for its discussion of this, and other potential pitfalls of significance tests.

Some standard packages for evaluating t-tests, F-tests, or the ANOVA also present analysis results in terms of critical values. Most of them do usually print the actual p value as well, along with the value of the test statistic and the degrees of freedom. Most researchers rightfully report the test statistics along with the actual significance levels: “we conclude that there is a significant difference in mathematical performance (t(88) = 2.499, p = 0.014)… .” Here, 88 gives the degrees of freedom, is the value of the t-statistic, and is of course the p-value.

Similar comments apply to the F-test, discussed in more detail below.

Assumptions

Strictly speaking, the t-test is only valid for normally distributed data where both populations have equal variance. However, the test is fairly robust to non-normal data [Box53]. You can verify that the sample variances are “equal enough” – that is, they could plausibly both be sampled observations from populations with the same variance, by using the F-test. The F-statistic

is distributed according to the F distribution with
degrees of freedom

Figure 6: The F distribution

In practice, the larger variance is usually put in the numerator, so . The test should still be two-tailed, so you should double the area under the right-hand tail³. In this situation, you want to check if you ƒshould accept the null hypothesis (that
) at a given significance level. If so, then you can go ahead and apply the t-test.

There is a variation of the t-tests for distributions of unequal variance, called Welch’s t-test [Wikc]. In this case, you are only checking if the means are equal, not that the distributions are the same.

The F-test for Analysis of Variance (ANOVA)

ANOVA is an extension of the difference of means test above to the casae of more than two populations. The null hypothesis in this case is that all the sample means are equal – or more strictly, that all the treatment groups are drawn from the same population.

The simplest version of the ANOVA is the one-way ANOVA, where there are treatment groups (populations) with subjects (or repetitions, or replications) each, for a total of subjects. Each population corresponds to a different single factor (a treatment or a condition: for example, a type of medicine, or a Star-Bellied Sneetch vs. a Plain-Bellied Sneetch vs. a Grinch). Two- or three- way ANOVAs correspond to varying two or three different factors combinatorially. For example, we could do a two-way ANOVA of Sneetch math performance by considering both the belly type and the gender of the Sneetchs.

Figure 7: Table for a Two-way ANOVA of Sneetch math performance

We will only discuss one-way ANOVA in this article, since that covers all the relevant ideas about calculating significance.

For a one-way ANOVA, we have the population means and variances . We can also calculate the overall mean , over the entire aggregate population.

The between-groups mean sum of squares, which is an estimate of the between-groups variance, is given by

(3)

is sometimes designated It is a measure of how the population means vary with respect to the grand mean.

The within-group mean sum of squares is an estimate of the within-group variance:

(4)

is sometimes designated . It is a measure of the “average population variance”.

Figure 8: Within-group and between-group variance

If the null hypothesis is true, then

is distributed according to the F distribution wiht
degrees of freedom.

Figure 9: p-value for the one-tailed F-test

That is, under the null hypothesis, the within-group and between-group variances should be about equal:
. If , then some of the treatment groups overlap other groups substantially, so practically speaking, one might as well accept the null hypothesis. Hence, a one-sided F test is good enough. As with the t-test, research papers usually give the value of the F statistic, the degrees of freedom, and the p-value: “
”. In this example, the test statistic value is 6.6, and it was evaluated against the F distribution with (2, 864) degrees of freedom, which means that
. The p-value is 0.0014.

Assumptions

Like the t-test, ANOVA assumes that the data is normally distributed with equal variances. According to Box [Box53], ANOVA is fairly robust to unequal variances when the population sizes are about the same, but you might want to check anyway. If all the populations are the same size (all the are the same), the easiest way to check for equality of variances is an F-test of the statistic
with degrees of freedom[Sac84]. In other cases, you can use Bartlett’s Test [Wika] or Levene’s Test [Wikb]. Bartlett’s test uses a test statistic that is distributed as the distribution, and Levene’s test uses one that is distributed as the F distribution. Levene’s test does not assume normally distributed data.

If the data are not normally distributed, or have unequal variance, often they can be transformed to a form that is closer to obeying the assumptions of ANOVA. The following table of transformations is based on [Sac84, p. 517], and other sources [Hor].

Figure 10: Table of Transformations

Jim Deacon from the University of Edinburgh lists some suggestions as well [Dea]. He also reminds us that running ANOVA on the transformed data will identify significant differences in the transformed data. This is not the same as saying there are significant differences in the original data!

Once the Null Hypothesis is Rejected

If you are able to reject the ANOVA null hypothesis, you will usually want to know which population means are significantly different from the rest. Often, in fact, you are primarily interested in which population had the highest mean. For example, if you are comparing the efficacy of a new medicine A against existing medicines B and C, you are probably not too concerned about whether B and C perform significantly differently from each other, only about whether A is significantly better than both.

If all you care about is whether the highest mean is significantly higher than the others, you can simply test where the statistic

falls on the Student-t distribution with degrees of freedom. Here, is the within-group variance, as calculated in Equation 4, and are the highest and second highest population means, is the total number of samples (
), and is the number of treatment groups.

This test is usually written

where
is the (two-sided) critical value for significance level and is the number of degrees of freedom to use. This quantity is called the least significant difference (LSD) between the highest and second highest means, and the test is usually called the LSD test.

If you want to test all the population differences for significance, (or test the highest value against all of the others explicitly) then you need to take some care with the LSD test. Remember that a significance level of means that with probability you will make a false positive error. To test all possible population differences is = ( choose ) comparisons, or comparisons, if you sort all the means in descending order and compare adjacent ones. Testing the highest mean against all the lower values is also comparisons. This means you have a
probability of making a false positive error. So if you want the overall significance level to be , each individual comparison should use a stricter significance threshold
.

A preferred way to compare multiple means for significance (once the ANOVA null hypothesis has been rejected) is to use a multiple range test [Dea] or Tukey’s method [oST06], rather than the LSD test. Tukey’s method tests all pairwise comparison simultaneously, and the multiple range test starts with the broadest range (the highest and the lowest means), and works its way in until significance is lost.

Conclusion

We’ve skimmed over many complications in this discussion. Hopefully, though, what we have gone over is enough to demystify much of the statistical discussion in research papers. Perhaps, it will demystify the output of standard ANOVA and t-test packages for you, as well.

Chong-ho Yu’s site [hY] gives a brief discussion of some of the issues that I’ve skimmed over. It also lists a few common non-parametric tests. These are tests that do not make assumptions about how the data is distributed, and so they may be more appropriate for data that is very non-normal, or for discrete data. They tend to have less power than parametric tests (that is, they have a lower true positive rate); so if the data is at all normal-like, parametric tests are preferred.

Significance tests are used in other applications beyond testing the difference in means or variances. They are used for testing whether events follow an expected distribution, for testing if there is a correlation between two variables, and for evaluating the coefficients of a regression analysis. We hope to cover some of these applications in future installments of this series.

Bibliography

Box53

G.E.P. Box, Non-normality and tests on variances, Biometrika 40 (1953), no. 3/4, 318-335.

Dea

Jim Deacon, A multiple range test for comparing means in an analysis of variance, http://www.biology.ed.ac.uk/research/groups/jdeacon/statistics/tress7.html.

FPP07

David Freedman, Robert Pisani, and Roger Purves, Statistics, 4th ed., W. W. Norton & Company, New York, 2007.

Hor

Rich Horsley, Transformations, http://www.ndsu.nodak.edu/ndsu/horsley/Transfrm.pdf, Class notes, Plant Sciences 724, North Dakota State University.

hY

Chong ho Yu, Parametric tests, http://www.creative-wisdom.com/teaching/WBI/parametric_test.shtml.

oST06

National Institute of Standards and Technology, Tukey’s method, NIST/SEMATECH e-Handbook of Statistical Methods, 2006, http://itl.nist.gov/div898/handbook/prc/section4/prc471.htm.

Sac84

Lothar Sachs, Applied statistics: A handbook of techniques, 2nd ed., Springer-Verlag, New York, 1984.

Wika

Wikipedia, Bartlett’s test, http://en.wikipedia.org/wiki/Bartlett's_test.

Wikb

—–, Levene’s test, http://en.wikipedia.org/wiki/Levene's_test.

Wikc

—–, Welch’s t test, http://en.wikipedia.org/wiki/Welch's_t_test.

Footnotes

…¹

Remember from the last installment that when you are estimating the mean of a distribution with unknown mean and unknown variance , the 95% confidence interval around your estimate is
. Intuitively speaking, Student’s distribution is what you get if you calculate confidence intervals using the estimated variance instead of the true but unknown variance . The distribution is stretched out compared to the normal distribution to reflect this increased uncertainty.

… positives²

In his textbook Statistics, Freedman tells an anecdote about a study that was published in the Journal of the AMA, claiming to demonstrate that cholesterol causes heart attacks. The treatment group that took a cholesterol reducing drug had “significantly fewer” heart attacks than the control group (
). A closer reading revealed that the researchers used a one-tailed test, which is equivalent to assuming that the treatment group was going to have fewer heart attacks. What if the drug had increased the risk of heart attack? The proper two-tailed significance of their results would have been
, which is higher than JAMA‘s strict significance threshold of 0.05. [FPP07, p. 550]

… tail³

The area to the right of with degrees of freedom is equal to the area to the left of , with degrees of freedom.

Related posts:

1. Statistics to English Translation, Part 2a: ’Significant’ Doesn’t Always Mean ’Important’
2. “I don’t think that means what you think it means;” Statistics to English Translation, Part 1: Accuracy Measures
3. Living in A Lognormal World

As always, a pdf version of this article is available as well.

Does too much salt cause high blood pressure, or doesn’t it? That debate has raged for decades, with a slew of studies finding “yes” and a slew of others finding “no.” Two new studies out today in the journal Hypertension tip the scales in favor of reducing sodium – particularly for those 1 in 4 Americans who have high blood pressure. One study found that reducing salt intake from 9,700 milligrams a day to 6,500 milligrams decreased blood pressure significantly in blacks, Asians, and whites who had untreated mild hypertension. Another study found that switching to a lower-salt diet helped lower blood pressure in folks with treatment-resistant hypertension.
- “10 salt shockers that could make hypertension worse,” U.S. News & World Report [Kot09]

“Great!” you think. “Who needs to spend money on high-blood pressure meds? I can just cut down my salt!” Well, maybe so, maybe not. To come to that conclusion, you need more information than you were given in that paragraph. What was the “significant” decrease in blood pressure? What was the “before” and the “after”? Does “significant” mean important, or useful? And why has there been so much controversy over this?

Let’s discuss the important points with an example.

Suppose that we wanted to test for a difference in intelligence between two groups, say Star-Bellied Sneetches and Plain-Bellied Sneetches². We take a group of 50 Star-Bellies and a group of 40 Plain-Bellies, and give them both a series of tests designed to measure their mathematical, linguistic, and problem-solving abilities. After evaluating the data, we conclude that there is “a significant difference in mathematical performance (t(88) = 2.499, p = 0.014) between the two groups”. The mean mathematics score of the Star-Bellies is 78, with a standard deviation of 7, and the mean mathematics score of the Plain-Bellies is 74, with a standard deviation of 8, for a difference of 4 points³.

Should we interpret this result to mean that Star-Bellied Sneetches are better than Plain-Bellied ones at math? It depends.

How Hypothesis Tests Work

The Sneetch example above and the blood-pressure study cited earlier are both examples of hypothesis tests. In hypothesis testing, researchers set their proposed hypothesis (that there is an effect or a relationship) against the null hypothesis that there is no effect or relationship. In this article, we consider proposed relationships of the form

The mean value of X measured for group A is different from the mean value of X measured for group B.

In this case, the null hypothesis is

The mean value of X is the same for groups A and B, and any difference observed in the data is only by observational chance.

In fact, we are actually testing the stricter null hypothesis:

The distribution of X is the same for groups A and B, and any difference observed is only by observational chance.

A and B are sometimes called treatment groups; this terminology comes from the original applications of hypothesis testing procedures, in agriculture and medicine. In the blood pressure study above, the treatment is daily salt intake. One group ingests about 9,700 milligrams of sodium a day, the other group about 6,500 milligrams a day. The question of interest is: does the difference in sodium intake make a difference in the average blood pressure of the two groups? The null hypothesis is “No.”

Significance

We call an observed difference significant – meaning that a difference as large as we observed is probably not by chance – if the the value is “high enough.” In the Sneetch example, is the significance level of the result. To interpret the p-value, suppose the null hypothesis is true: there is truly no difference between Star-Bellied math scores and Plain-Bellied math scores. If this is so, then there is only a 0.014 (1.4%) chance that the difference in the average scores of the two groups will be 4 points or larger. In other words, if the null hypothesis is true, and we administer this same test to different groups of 50 Star-Bellies and 40 Plain-Bellies a hundred times, then the difference in scores will be 4 points or more only about once or twice.

We interpret the fact that we have seen a difference that should be rare to be evidence that the null hypothesis isn’t true. So we reject the null hypothesis and say that there is a “significant difference” in the performance of the two groups. Alternatively, we could say that Star-Bellied Sneetches performed “significantly better” than Plain-Bellied Sneetches on the math test.

Effect Size

Four points (or about a 5% difference) is the effect size of the comparison. The effect size represents what might be called the “practical significance” of the result. In general, the larger the effect size, the better. In this example, Star-Bellies might truly outperform Plain-Bellies by about four points on average, but if we were to examine the relationship between math scores and real-life math performance (say, how well college-attending Sneetches do in their math and science courses), we might discover that it takes a test score difference of ten points or more to reliably predict which Sneetches will do better. In that case, a four point average difference would not be a practical difference.

Evaluating a Result

When evaluating a result, you should look both for its significance and its effect size. In practice, researchers usually consider a finding to be significant if
. This is actually a pretty large ; it means even if the null hypothesis is true, you would still observe a difference as large as the one that you observed about five times out of every one hundred trials. In fact, Sachs noted that
used to be the commonly used threshold for significance ([Sac84, p. 114]).

Sometimes results are reported using an asterisk convention: (*) means
, (**) means
, and (***) means
. Hopefully, the actual significance level is reported (it isn’t always), as well as the actual effect size (it isn’t always).

The effect size in medical studies is often reported in the popular press with statements like “those who abstained from coffee had triple the risk of contracting colon cancer compared to those who drank three or more cups a day.” Does that mean that all confirmed Lapsang Souchong drinkers and the uncaffeinated should run out and learn to embrace Starbucks? Well, no. First of all, ask yourself: what is the baseline risk of colon cancer? If abstaining from coffee triples the risk from 0.01% to 0.03%, well, it probably isn’t worth worrying about. On the other hand, if the risk triples from 5% to 15%, perhaps that is a reason to take up espressos. You should also see who were the subjects of the study, and how similar they are to you. Suppose the study was done on Caucasian males in the U.S., ages 55-65, with no family history of colon cancer. If you are a young white American male, it’s possible that this study says something about your future health. If you are female or non-Caucasian or not living in the U.S., the finding may or may not be relevant to you. It depends on the mechanism that drives the relationship, and whether or not it applies to you as well as to the subjects of the study.

“Significant” is not the same as “Important”

With a large sample, even a small difference can be “statistically significant”… . This doesn’t necessarily make it important. Conversely, an important difference may not be statistically significant if the sample size is too small.
- Freedman, Pisani and Purves, Statistics [FPP07, p. 550]

The ability of a study to detect a significant difference depends almost entirely on its size. When a researcher designs a study, she has to decide how much risk of error – and what type of error – she is willing to tolerate.

How big a risk [of inventing a difference] between two indistinguishable treatments are we willing to put up with? This risk is known as the significance level . [Sac84, p. 214]

is the probability of rejecting a null hypothesis that should be accepted. This is a Type I error (a false positive). enters the design of the study as the threshold for p-values that the researcher will accept as significant.

How big a risk do we allow of missing a substantial difference between two treatments? … This risk is called . [Sac84, p. 214]

is the probability of accepting a null hypothesis that should have been rejected. This is a Type II error (a false negative). The quantity is known as the power of the test: the probability that the test will correctly reject the null hypothesis when the alternative hypothesis is true.

How small a difference should still be recognized as significant? This difference is called . [Sac84, p. 214]

is the minimum effect size that we are willing to consider “practically significant.”

It is important to consider all three of , , and when determining an appropriate sample size for a trial. The power of a test and the significance of a result both increase as the sample size increases. So if is not specified, any difference can appear significant, with a large enough , even if the difference is really by chance.

The Central Limit Theorem

To see why the above statement is true, we need a few more facts about estimating the mean. Suppose we have a random variable that is normally (or nearly normally) distributed, with a true mean and (unknown) variance . You want to estimate by drawing samples; the sample mean gives you an estimate of . According to the Central Limit Theorem, if you were to repeat this experiment over and over again, you would see that the estimated has a normal distribution, with mean and variance
. So is a good estimate of , one that improves with a larger sample size .

Another fact about normal distributions is that a little over 95% of the probability mass is within standard deviations of the mean. So, for a single experiment, we can reason that the true mean is in the interval
with 95% probability⁴.

Figure 1: Confidence bounds on the estimate of for different values of

So, as gets larger, we zoom in on ⁵.

Now, back to the problem of checking for the difference of means. We’ll take samples from population and from population . Let’s assume for now that the variances are equal.

Figure 2: Confidence bounds overlap; means may not be truly different

With 95% probability,
, and
. If
is small compared to
, then the two confidence intervals overlap substantially, and we cannot reject the null hypothesis that
.

If, on the other hand,
is wide compared to
:

Figure 3: Confidence bounds don’t overlap; means are significantly different

then the confidence intervals are well separated, and we can reject the null hypothesis.

So , the minimum significant distance – the “resolution” of the experiment – is about the distance when the two confidence intervals touch:
, if our desired significance level is 0.05.

Figure 4: Minimum significant distance for a given sample size

If is too large, the experiment may be unable to detect important differences because the confidence intervals overlap too soon. This means that the sample size was too small (the test didn’t have enough power), and the experiment should be repeated with a larger test population.

If is too small, then the experiment will potentially detect statistically significant differences that are, for all practical intents and purposes, meaningless. To go back to the Sneetch example, if the math exam has one hundred questions, then an effect size of two points would correspond to one group answering two additional questions correctly, on average. Practically speaking, that’s probably not a very big difference. But if we made the experiment big enough, about 250 Sneetches in each group, it would be a statistically significant difference, to the 0.05 level. In theory, we could even make a difference of less than one point statistically significant! That is why knowing the effect size of a significant result is important.

“Significant” is not the same as “True”

The power and significance level of a test play similar roles to the sensitivity and specificity of a diagnostic test. You’ll remember from Part 1 of this series⁶that sensitivity and specificity are properties of the test, not how the test performs in a given population. To know the practical accuracy of a screening test, you must know the underlying prevalence of the condition that it is screening for. If it is crucial that the screening not miss any positive cases, then the test will be designed to be highly sensitive, possibly at the cost of specificity. In that case, the test will tend to have a high false positive rate if the condition is relatively rare. And yet, this same screening test will have a lower overall false positive rate when used in a population where the condition is more prevalent.

The same is true for hypothesis tests. The probability that a statistically significant result is actually true depends on the underlying probability that results “of that type” tend to be true in the domain of study. It also depends on whether the researcher was trying to minimize the chance of a false positive error, or a false negative error.

You should also be careful interpreting the results of exploratory work, where the researchers have run a series of several different studies, but only highlight the “significant” ones. Running twenty experiments and having one of them return a significant result to the level is actually not significant at all.

John Ioannides discusses these points (and a few others) in his 2005 essay “Why Most Published Research Findings are False”[Ioa05]. The essay made a few waves at the time of its publication, and it is still available online. We recommend that you read it, along with the 2007 followup article by Moonesinghe, et.al [MKJ07]. Now that you’ve read the first two installments of the Statistics to English translation, both essays should be a breeze!

Some Points to Remember

• “Significant” is a statistical statement that an observed relationship is unlikely to be by chance. It is not an necessarily a statement about the magnitude or the importance (or the truth!) of the relationship.
• Knowing the effect size of a significant result will help you decide if the relationship is “practically significant.”
• With a large enough sample size, any difference in means can appear significant, even when it is by chance.

You now have a general idea what a “statistically significant result” is. The next installment will go into a little more technical detail of how significance is calculated. You should read that installment if you want to decipher statements in research papers like “
” — or if you are simply curious.

Bibliography

FPP07

David Freedman, Robert Pisani, and Roger Purves, Statistics, 4th ed., W. W. Norton & Company, New York, 2007.

Ioa05

John P. A. Ioannidis, Why most published research findings are false, PLoS Med 2 (2005), no. 8, e124, Available as http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0020124.

Kot09

Deborah Kotz, 10 salt shockers that could make hypertension worse, U.S. News & World Report (2009), Online as http://health.usnews.com/articles/health/heart/2009/07/20/10-salt-shockers-that-could-make-hypertension-worse.html.

MKJ07

Ramal Moonesinghe, Muin J Khoury, and A. Cecile J. W Janssens, Most published research findings are false — but a little replication goes a long way, PLoS Med 4 (2007), no. 2, e28, Available as http://www.plosmedicine.org/article/info%3Adoi%2F10.1371%2Fjournal.pmed.0040028.

Sac84

Lothar Sachs, Applied statistics: A handbook of techniques, 2nd ed., Springer-Verlag, New York, 1984.

SS99

Murray R. Spiegel and Larry J. Stephens, Schaum’s outline of statistics, 4th ed., McGraw-Hill, New York, 1999.

Footnotes

… series¹

http://www.win-vector.com/blog/category/statistics-to-english-translation/

… Sneetches²

“The Sneetchs,” from The Sneetches and Other Stories by Dr. Seuss.
http://www.youtube.com/watch?v=Ln3V0HgW4eM
and http://www.youtube.com/watch?v=s0LgMpfLD1Y

… points³

This example is based on Exercise 10.17 in [SS99]; the original exercise did not, unfortunately, involve Sneetches.

… probability⁴

The correct way to state this is that for a given (unknown) , the estimate falls in the interval
just over 95% of the time. This gets awkward to reason about. Luckily, symmetry arguments let us center the appropriate confidence interval around instead.

…⁵

Of course, we don’t actually know , so we don’t know exactly how fast we zoom in. That doesn’t affect our argument, though, since only changes

… series⁶

http://www.win-vector.com/blog/2009/11/i-dont-think-that-means-what-you-think-it-means-statistics-to-english-translation-part-1-accuracy-measures/

Related posts:

1. Statistics to English Translation, Part 2b: Calculating Significance
2. “I don’t think that means what you think it means;” Statistics to English Translation, Part 1: Accuracy Measures
3. R annoyances

The “Statistics to English Translation” series is a new set of articles that we will be posting from time to time, as an attempt to bridge the language gaps. Our goal is to increase statistical literacy: we hope that you will find it easier to read and understand the statistical results in research papers, even if you can’t replicate the analyses. We also hope that you will be able to read popular media accounts of statistical and scientific results more critically, and to recognize common misunderstandings when they occur.

The first installment discusses some different accuracy measures that are commonly used in various research communities, and how they are related to each other. There is also a more legible PDF version of the article here.

The Basics

In informal language and in popular press articles, “accuracy” is often discussed as if it were a one-dimensional property of a diagnostic test or a classifier.

In general though, a single number is not enough. A test or classifier should detect what’s interesting, and ignore what’s not. How well it accomplishes these two tasks is related to the two kinds of mistakes that a test or classifier can make: false negatives, and false positives.

For a classification task, positive means that an instance is labeled as belonging to the class of interest: we may want to automatically gather all news articles about Microsoft out of a news feed, or identify fraudulent credit card transactions. For a screening test, positive means that the test detects whatever it was designed to look for: an HIV test detects the presence of human immunodeficiency virus, for example, while an allergy test detects the presence of an allergic reaction. A negative is obviously the opposite of a positive.

A false positive is concluding that something is positive when it is not. False positives are sometimes called Type I errors. A false negative is concluding that something is negative when it is not. False negatives are sometimes called Type II errors. The terms “Type I error” and “Type II error” are not terribly mnemonic, but they are commonly used, and therefore worth knowing.

For binary classification or binary test procedures, the False Positive Rate, , is the fraction of negative instances that are erroneously misclassified as positive.

(1)

Likewise, the False Negative Rate, , is the fraction of positive instances that are erroneously misclassified as negative.

(2)

The True Positive Rate, , is the fraction of positive instances that are correctly identified as such. It follows from the Definition 2 above that
.

The True Negative Rate, , is the fraction of negative instances that are correctly identified as such. It follows from the Definition 1 above that
.

Sensitivity and Specificity

The terms sensitivity and specificity generally refer to diagnostic or screening procedures, such as an HIV or allergy tests. The sensitivity of a test is its true positive rate; the specificity is its true negative rate, although it can be more intuitive to think of specificity as the complement of the false positive rate: Specificity =
.

The Wikipedia entry on Sensitivity and Specificity [Wik] uses a nice example to illustrate the difference: think of a drug-sniffing dog as a screening test for illicit drugs. If the dog’s nose is highly sensitive to the smell of drugs, then it will detect all the hidden packets of drugs; if it is less sensitive, then it will fail to detect some of the packets. At the same time, the dog should react specifically to drugs, and not, say, jambalaya or doggie biscuits. If the dog is highly specific in its reactions, it will only react to drugs; if it is less specific, then it will react to the occasional care package of yummy home cooking from Mom.

Screening tests may trade off specificity for sensitivity (and vice-versa). To go back to our drug-sniffing example, we might treat every suitcase and bag that comes through the airport as if it contained drugs; this procedure is perfectly sensitive (it will detect every packet of drugs, for sure), but not specific at all. Or, we might assume that no one is carrying drugs. This is perfectly specific (we will never make a false accusation), but not sensitive at all.

A more realistic example, inspired by a discussion of mandatory AIDS testing by Joshua Rosenau [Ros06], is the use of the ELISA screening test to detect HIV-infected blood donations. The ELISA test is designed to be very sensitive: it detects 99.7% of the cases of HIV-infection, which gives a false negative rate of
. On the other hand, it is not very specific: it has a 1.9% false positive rate¹. If you assume that the incidence of HIV-positive individuals in the general population is about 448 out of every 100,000 people [Hig08], then a positive test result is correct only about 19% of the time: one case of true infection out of every five positives. This error rate may be appropriate for screening blood donations, since it is better to discard four perfectly good pints of blood, “just in case”, than to allow a pint of HIV-infected blood into the blood bank. But it is not appropriate to assume that all five of those poor blood donors are HIV-positive, without followup tests to increase the specificity of the screening procedure.

Sensitivity, Specificity, and Prevalence

The example above brings up an important point. Sensitivity and specificity are properties of the test itself, not how the test performs in a given population. The absolute accuracy (as the term is commonly understood) of a screening test will change, depending on the prevalence of the condition that the test is screening for.

Let’s imagine the ELISA test described above as an HIV-screening daemon, who uses two coins to generate uncertainty. When the daemon is shown a pint of infected blood, she flips an unfair quarter. If the quarter comes up heads (which it does 3 times out of every 1000 flips), then she lies and says the blood is uninfected, otherwise she tells the truth. When the daemon is shown a pint of uninfected blood, she flips a silver dollar that comes up heads about 2 times out of every 100 flips. If the silver dollar comes up heads, she lies and says the blood is infected, or else she tells the truth. The quarter and the silver dollar encode ELISA’s sensitivity and specificity, respectively.

Figure 1: The ELISA daemon screening an uninfected pint of blood

Suppose ELISA looks at the blood of 1000 people a day, drawn from the general population. We can expect that about 5 of them are truly infected. That means that ELISA flips her silver dollar 995 times; it will come up heads about 20 times. That’s about twenty false positives a day. She’ll flip the quarter about 5 times, and with high probability, won’t ever see a head. That’s near zero false negatives a day. In total, ELISA will read positive for about 25 pints of blood every day, and she will be wrong for 80% of those cases.

But suppose ELISA looks at the blood of 1000 people from a high-risk population, where one out of four people are infected. Then ELISA flips her silver dollar about 750 times, and it will come up heads about 15 times: 15 false positives. She’ll also flip the quarter 250 times; the coin just might come up heads one time. Let’s say it does. Then ELISA will read positive for 249+15 = 264 pints of blood, and she’ll be wrong for only about 6% of those cases – plus that case of infection that she missed.

Same test, same sensitivity and specificity, but different overall accuracy. The percentage of positives that are actually true positives in a given population is called the positive predictive value ( ) of the test within that population; it is the probability for that population that a positive test result correctly predicts a positive instance.

(3)

where is the probability of a positive instance, or in other words the prevalence of the condition in the population. is the probability of a negative instance, and of course
.²

Likelihood Ratios

Likelihood ratios are another measure of diagnostic test accuracy. The positive likelihood ratio is the true positive rate over the false positive rate:
. For our example ELISA test, the positive likelihood ratio is 0.997/0.019 = 52.47. The negative likelihood ratio is the false negative rate over the true negative rate,
. For our ELISA example, the negative likelihood ratio is 0.003/.981 = 0.003058.

Likelihood ratios are a property of the screening test, independent of the prevalence of the condition in the population. If you know the odds of infection for the population of interest,
, then you can calculate the posterior odds of infection for someone who has tested positive:

and the posterior odds of infection for someone who has tested negative:

It’s been argued that likelihood ratios make it easier for non-statistically-minded practitioners to interpret the results of a test than sensitivity and specificity do [JGS94]. It’s also been argued the other way [PSBtR05]. Which framework makes more sense depends on if you prefer thinking in odds or probabilities. In either case you should be leery of “guidelines” of the sort: “ indicates large and often conclusive increase in the likelihood of the disease.” There is certainly a large increase in the posterior likelihood of infection if , but as the ELISA coin-flipping example should have made clear, this posterior likelihood can still be quite small, if the disease is sufficiently rare.

I occasionally see something called the diagnostic odds ratio. It was developed as “a single indicator of test performance,” and I’ve seen it described as “the odds of the true positive rate divided by the odds of the false positive rate” [HC07]. I could give you the actual formula here, but frankly – it makes no sense. The whole point of having two measures for accuracy is that one is not enough, and if you absolutely must have one number, you are better off using something like the measure that we describe in the next section.

Precision and Recall

Precision and recall are similar (but not identical) to sensitivity and specificity. The measures are popular in the information retrieval and machine learning communities.

Recall is the same as sensitivity, or the true positive rate: the number of true examples correctly classified as such. Precision is defined as the fraction of instances classified as positive that really are positive:

   precision

This is not the same as specificity; it is the same as the positive predictive value, and is a joint property of the classifier and the population that it was evaluated on.

Information retrieval research concerns itself with efficient discovery of relevant documents from document collections, and that domain motivates the definitions of precision and recall. A library patron queries the library catalog for books on a given topic; the catalog’s search engine should return all of the books relevant to her query, and only those books. Recall is a measure of how well the search delivers “all of the relevant books”, and precision is a measure of how well it delivers “only the relevant books”. If recall is poor, then the patron will miss finding many relevant books; if precision is poor, then she will be inundated with a bunch of book suggestions that have nothing to do with her search.

As with diagnostic procedures, classifiers may trade precision for recall, and vice-versa. Suppose our library patron is looking for novels about vampires. She could request all novels with the word “vampire” in the title. This search would have almost perfect precision, since presumably a novel with the word “vampire” in the title is going to be about vampires. It would not have perfect recall, since many novels about vampires – like Dracula, or the books from the Twilight series – don’t announce themselves quite that blatantly. Now suppose she is only interested in novels from Anne Rice’s Vampire Chronicles series. She could request all novels authored by Anne Rice. This search would have perfect recall, but not perfect precision, since Ms. Rice did in fact write several novels that are not about vampires.

These examples show that neither high precision nor high recall guarantee a useful classifier. It is the tension between achieving high precision and high recall that leads to good classifiers.

As we discussed above, the primary difference between precision and specificity is that precision is a property of the algorithm and the population. One could argue that precision is a more appropriate measure than specificity for many classification and machine learning tasks, especially those related to text or natural language. The fundamental assumption, after all, is that such algorithms are trained on data that is representative of the population that the classifier will be deployed in.

If you insist: Single Score Measures

There is another measure called , the harmonic mean of precision and recall:

is near one when both precision and recall are high, and near zero when they are both low. It is a convenient single score to characterize overall accuracy, especially for comparing the performance of different classifiers.

Using to compare classifiers assumes that precision and recall are equally important for the application. If one criterion is more important than the other, then one can also use the weighted geometric mean:

precision   recall   precision + recall

describes how much more important recall is than precision: use if recall is twice as important as precision, if precision is twice as important as recall.

It is still better to have separate target goals for precision and recall that a candidate classifier must meet. Still, and are found in the literature, so they are presented here.

ROC Curves

Not all diagnostic tests or classifiers return a simple “yes-or-no” answer. In fact, most probably don’t. Generally, a classification or diagnostic procedure will return a score along a continuum; ideally, the positive instances score towards one end of the scale, and the negative examples towards the other end. It is up to the scientist or the analyst to set a threshold on that score that separates what is considered a positive result from what is considered a negative result. The Receiver Operating Characteristic Curve, or ROC Curve, is a tool that helps set the best threshold.

Figure 2: Plot of score distributions for positive and negative instances (Class 10 is positive)

Suppose we are trying to classify a set of instances into one of two classes, positive and negative³. We’ve gathered a test set of representative samples, and we’ve developed a scoring procedure to try to separate them. Positives tend to score on the high end of the scale, negatives toward the low end. We want to pick a threshold value.

Figure 2 shows what happens when score the test set. We can see that the scores of the positive instances (class 10) are in a cluster centered just above 7, and the scores of the negatives (class 0) are in a cluster centered near 5. Still, there is an interval where the two clusters overlap substantially. If we pick a threshold to the right of that interval (say, ), almost everything that scores greater than will be truly positive (high precision/specificity), but we miss a lot of positives, too (low recall/sensitivity). If we pick a threshold to the left of that interval (say ), we will catch almost all the positives (high recall/sensitivity), but we will also pick up a lot of negatives (low specificity/precision). So we want the threshold to be somewhere in the overlap interval, but where?

Figure 3: ROC Curve corresponding to Figure 2. Selected thresholds are marked on the curve.

ROC curves plot the false positive rate on the x-axis and the true positive rate on the y-axis, as we vary the threshold. The point corresponds to rejecting everything; the point corresponds to accepting everything. The ideal point is : accept all positive instances and reject all negative instances. The line corresponds to random guessing: that is, a procedure that assigns each instance a score uniformly drawn from (in this example) the interval without even checking if the instance is positive or negative.

The ROC curve represents the tradeoff between true positives and false positives that we make as we increase the threshold from accepting everything to rejecting everything. Figure 3 gives the ROC curve for our example, with a few example thresholds marked on the curve.

The area between the ROC curve and the line can be considered a measure of accuracy; the smaller that area, the more the scoring procedure is like random guessing. The larger the area, the better separated the two classes are. We can use the curve to help us decide how to set a threshold that will give us the most acceptable tradeoff between true positives and false positives. For this example, we would probably want to select a threshold somewhere between and .

In Conclusion

Some points to remember:

• Classifier and diagnostic test performance are not one-dimensional.
• Different fields use different (but related) measures of accuracy.
• Classifier and diagnostic test performance depend on the relative cost of Type I and Type II errors, as well as on the proportion of positive and negative instances in the population of interest.

Bibliography

HC07

Childrens Mercy Hospitals and Clinics, Stats: Meta-analysis for a diagnostic test, http://www.childrens-mercy.org/stats/model/diagnostic.asp, 2007.

Hig08

Liz Highleyman, CDC updates estimates of HIV prevalence in the United States, http://www.hivandhepatitis.com/recent/2008/100708_a.html, 2008.

JGS94

R. Jaeschke, GH Guyatt, and DL Sackett, Users’ guides to the medical literature. III. How to use an article about a diagnostic test. B. What are the results and will they help me in caring for my patients? The evidence-based medicine working group, JAMA 271 (1994), no. 6, 703-707.

Org04

World Health Organization, HIV assays: Operational characteristics (phase 1); Report 15 antigen/antibody ELISAs, http://whqlibdoc.who.int/publications/2004/9241592370.pdf, 2004.

PSBtR05

MA Puhan, J. Steurer, LM Bachmann, and G. ter Riet, “A randomized trial of ways to describe test accuracy: the effect on physicians’ post-test probability estimates”, Annals of Internal Medicine 143 (2005), no. 3, 184-âÄì189.

Ros06

Joshua Rosenau, AIDS testing, http://scienceblogs.com/tfk/2006/08/aids_testing.php, 2006.

Wik

Wikipedia, Sensitivity and specificity, http://en.wikipedia.org/wiki/Sensitivity_and_specificity).

Appendix: Glossary of Accuracy Terms

Basic Terms

Accuracy

Type I error

False Positive: to conclude something is a positive instance when it is not.

Type II error

False Negative: to conclude something is a negative instance when it is not.

False Positive Rate ( )

The fraction of negative instances that are erroneously misclassified as positive.

False Negative Rate ( )

The fraction of positive instances that are erroneously misclassified as negative.

True Positive Rate ( )

The fraction of positive instances that are correctly identified as such.
.

True Negative Rate ( )

The fraction of negative instances that are correctly identified as such.
.

Prevalence

The proportion of positive instances in the population, or the probability that someone drawn from the population at random in a positive instance. is the probability of drawing a negative instance from the population at random. The odds of a positive is the ratio of to .

Accuracy Terms

Terms to describe the accuracy of diagnostic tests are conventionally given in terms of sensitivity and specificity. They have been rephrased here in terms of true positive rate, false positive rate, etc., for clarity.

Single score measure of accuracy. The harmonic mean of precision and recall.

is near 1 for high accuracy, near 0 for low accuracy. Also see Precision, Recall.

Single score measure of accuracy when recall is twice as important as precision. Also see , Precision, Recall.

Single score measure of accuracy when precision is twice as important as recall. Also see , Precision, Recall.

Likelihood-ratio (negative)

. In diagnostic screening, used for calculating the posterior odds of a true positive for a subject who has tested positive:

Likelihood-ratio (positive)

. In diagnostic screening, used for calculating the posterior odds of a positive for a subject who has tested negative:

Positive Predictive Value

Probability (with respect to a specific assumed prevalence rate) that a positive result from a diagnostic or screening procedure is a true positive. Same as precision.

Also see Precision

Precision

In information retrieval, the fraction of returned documents that are actually relevant to the query. In classification, the fraction of all instances classified as class A that are truly in class A. The same as Positive Predictive Value.

   precision

Also see Positive Predictive Value

Recall

In information retrieval, the fraction of relevant documents that are returned by the query. In classification, the fraction of all true instances of class A that are classified into class A. The same as sensitivity.

   recall

Also see Sensitivity

ROC Curve

Plot of true positive rate versus false positive rate for a diagnostic test or binary classifier, as the decision threshold is varied.

Sensitivity

The true positive rate of a diagnostic or screening procedure. Also see Recall.

Specificity

The true negative rate
(or the complement of the false positive rate) of a diagnostic or screening procedure.

Footnotes

… rate¹

The ELISA sensitivity and specificity numbers are from WHO’s report on the operational characteristics of HIV Assays [Org04, p. 18], using the lower bounds of the confidence interval. They are slightly different from the numbers Rosenau uses

….²

The definition of is conventionally given in terms of sensitivity and specificity (similarly for the likelihood ratios discussed in the following section). The definitions in this article are given in terms of false positive rate, etc., since that is clearer for people reading outside their discipline.

… negative³

We are using a classifier in our example, but a diagnostic test would work the same way.

Related posts:

1. Statistics to English Translation, Part 2a: ’Significant’ Doesn’t Always Mean ’Important’
2. Statistics to English Translation, Part 2b: Calculating Significance
3. Living in A Lognormal World

When a graph is made, quantitative and categorical information is encoded by a display method. Then the information is visually decoded. This visual perception is a vital link. No matter how clever the choice of the information, and no matter how technologically impressive the encoding, a visualization fails if the decoding fails. Some display methods lead to efficient, accurate decoding, and others lead to inefficient, inaccurate decoding. It is only through scientific study of visual perception that informed judgments can be made about display methods. The display methods of Elements rest on a foundation of scientific enquiry.

— from the preface of The Elements of Graphing Data

A revised edition of The Elements of Graphing Data was published in 1994, along with a companion volume, Visualizing Data, which is oriented towards the implementation and technical details of different graphing techniques. I highly recommend The Elements of Graphing Data as a guidebook for creating graphs, as well as for its excellent survey of several useful techniques. Cleveland, along with other colleagues at Bell Labs, developed the Trellis display system, a framework for the visualization of multivariable databases, using the ideas developed in his texts. Trellis, in turn, influenced Deepayan Sarkar’s Lattice graphics system for R. Lattice implements many of Cleveland’s ideas, and I also recommend Sarkar’s Lattice manual if you do data visualization in R.

It’s important to note here that Cleveland writes for researchers and decision-makers who use graphs to analyze data, or to convey scientific results to colleagues in an (ideally) objective manner. This distinguishes him from Darrell Huff, whose 1954 How to Lie with Statistics considered the use of graphs (and statistics in general) as rhetorical devices for convincing others of one’s point of view. Hence, some of Cleveland’s recommendations and guidelines actually contradict Huff’s. ¹

Edward Tufte also explored the idea that the choice of graphical display should be influenced by the viewer’s cognitive processes, in his 1990 book Envisioning Information. Tufte tends to be more broadly concerned with the gestalt of a graph, beyond its use as an analysis tool; he is also more concerned than Cleveland is with aesthetic considerations.

Cleveland’s philosophy might be summarized as: minimize the mental gymnastics that the viewer must go through to understand the graph. This leads to some obvious advice: avoid clutter and occlusion, make graphing symbols or color-coding unambiguous, use scale-lines on all four sides of the graph, and so on. It also leads to advice that perhaps should be as obvious, but isn’t: make the aspect of the data that you want to analyze as clear as possible. But what does this mean in practice?

Make important differences large enough to perceive

Weber’s Law is a well known observation from the psychophysics literature, which states that the “just noticeable” change in a stimulus is a constant ratio of the original stimulus. Put another way, people are only capable of detecting a change in a stimulus that is greater than a certain percentage k of the original stimulus. Here, “stimulus” can refer to any perceivable physical quantity: weight, intensity, length, orientation. The percentage k will vary with stimulus, and with observer.

Figure 1: From Cleveland, The Elements of Graphing Data

Figure 1 shows the application of Weber’s law to lengths. The bars A and B are of different lengths, but the difference is such a small fraction of the “base” length (say, A’s length, to be specific) that is difficult to tell whether or not they are different, or which is longer. On the right, the bars have been embedded in frames of identical length, and now it is easy to see that B is longer. Why? Because the difference in lengths of the white intervals is a much larger percentage of the white “base” length (say the white A interval). It is easy to see that the white B interval is shorter than the white A interval, and therefore, the black B interval is longer than the black A interval.

The moral is that you always want the viewer to be estimating changes or differences with respect to a short base length. You can do this with reference grids, as demonstrated below.

From Cleveland, The Elements of Graphing Data

Figure 2	Figure 3

Figure 2 shows eight curves. Which one dips to the lowest minimum? Are the high curves approaching the same value, and which one is rising the fastest? Are the low curves dipping to the same minimum? Are they going to the same steady state? Figure 3 shows the same curves, graphed with identical reference grids. The grids shorten the base lengths that are being compared, and it is now much easier to compare highs, lows, and steady state behavior.

But wouldn’t it be better to compare the graphs by superposing them? For two or three curves, perhaps. But in this case, eight curves can clutter the graph, and use up the symbol or color space, making it difficult to distinguish the different datasets — increasing the mental gymnastics.

Reference grids are useful even for a single curve, especially one with slowly varying segments, such as these graphs have. The reference grid makes it easier to answer questions like: does the process return to the initial state, or to a different steady state? Has the process reached steady state, or is it still growing?

Make important shape changes large enough to perceive: Banking to 45 degrees.

The aspect ratio of a graph is important when trying to understand shape. Rate of change information is encoded in the slope of the curve, which the viewer estimates by changes in the orientation of the local tangents at each point of the graph. Weber’s Law tells us that very small changes in this orientation will be difficult to detect. For a given (physical) curve, the local orientation changes will be dependent on the aspect ratio of its graphical presentation, as shown (to an exaggerated degree) in Figure 4. Here, the same curve (two line segments) is plotted at three different aspect ratios, one that centers the graph at 45 degrees, one that forces the curve to be nearly vertical, and another that forces it to be nearly horizontal. In the last two cases, the change in orientation of the two line segments is so small as to be nearly undetectable.

Figure 4: From Cleveland

For two line segments with positive, unequal slopes, a simple geometric argument shows that their absolute difference in orientation is maximized by the aspect ratio that sets their average orientation to 45 degrees (the first graph in Figure 4). Empirical studies by Cleveland and others have indeed verified that a viewer’s ability to judge the relative slopes of line segments on a graph is maximized when the absolute values of the orientations of the segments are centered on 45 degrees.

This result leads to a technique called Banking to 45, whereby the aspect ratio of the graph is chosen so that the average slope of the entire graph is 45 degrees. The details are discussed in Cleveland, and many of the plots in R’s Lattice package also have an option to bank the graph to 45 degrees.

This deliberate exaggeration of slope is something that Darrell Huff deplores. In How to Lie with Statistics, Huff refers to these graphs as “gee-whiz” graphs — and in the context of his discussion of statistics as rhetoric, they are:

Figure 5: From Huff, How to Lie With Statistics

To insist that a graph should always include a zero line and that units be in proportion may be good advice from a rhetorical perspective; but it is poor advice if the purpose of the graph is data analysis. As Figure 6 below demonstrates, we can lose resolution if we always insist on including the zero. Does the trend line in the left graph increase linearly, superlinearly, or sublinearly? The convexity of the curve is more apparent when it is banked to 45, as on the right. Assuming that the scientist reads the axis and is cognizant of the actual magnitude changes involved, the graph on the right conveys more information.

Figure 6: From Cleveland

Make sure all the data is equally well resolved.

It is quite common for positive data — word frequencies, populations, price distributions, just to name a few examples — to be skewed: most of the data is bunched towards low values, the rest of it is spread out on a very long tail. This long tail squashes the majority of the data into a tiny interval of a very narrow dynamic range, as in Figure 7, making it difficult to evaluate the data.

Figure 7: Long-tailed distribution of purchase sizes

Figure 8: Distribution of log(purchase size)

Imagine that Figure 7 represents the distribution of average purchase size across an online merchant’s customers: average purchase size is plotted on the x-axis, and the y-axis represents the fraction of the total customer population whose average purchase size is a given value (the area under the graph integrates to one). According to this graph, most customers make fairly small purchases on average, but there is a long tail of big spenders trailing out into the range of several thousand dollars. Obviously, one would like a little more resolution on the big spike of customers near zero. One could simply “zoom in” on this range, by chopping off some long chunk of the tail, but you may potentially lose sight of some global patterns in the data by doing so.

Graphing the distribution of log(purchase size) enables you to increase the resolution near zero, while preserving the global view. Figure 8 shows the distribution of log(purchase size), revealing two spending populations: a population of high spenders who tend to make purchases in the $3000 range (in log space), and another population whose purchases are centered (in log space) around $60. The existence of these two distinct populations is not apparent in the original graph.

Notice that Figure 8 has two x-axis scales: the top axis is marked in log units, while the bottom axis is marked in absolute dollars, spaced on a log scale. This accords with the principle of minimizing mental gymnastics, since the viewer of the graph will typically be concerned about prices in dollars, not log dollars. In fact, it would have been better yet to have plotted the distribution of log₂ or log₁₀ of the data; the former would allow us to see at a glance the doubling of price ranges, the latter to see price changes in factors of ten.

Figure 9: The 14 most abundant elements in meteorites. From Cleveland

Figure 9 shows another example: the fourteen most abundant elements in meteorites, specifically the average percent of each of the elements. If we graph the percentages directly, as on the left, we cannot easily distinguish the differences in the elements from aluminum on down. Graphing log₂ of the percentages, as on the right, improves the resolution. Again, we have two x-axes on the graph of the log data.

If you want to analyze the difference between two processes, then graph the difference, not the processes (or graph both).

Suppose that we are comparing the two processes f1 and f2 that are shown in Figure 10. As x increases, the two processes appear to be approaching each other — that is, the difference between the two seems to be decreasing. In reality, the difference between the two is constant: f2 = f1+1.

Figure 10: The illusion of convergence

Figure 11: British Imports and Exports. From Cleveland

It turns out that people are good at perceiving the perpendicular difference between two curves, but not the differences in height, which is what we are actually interested in here. When we try to infer the differences from the process graph, we may not only miss key information, we may actually draw incorrect conclusions.

A less toy example is given in Figure 11. Here the imports to and exports from England are graphed over the first 80 years of the 18th century. In the difference graph on the bottom, we can see a local peak in (imports-exports) just after 1760; this is not obvious from simply comparing the two processes (top graph).

If you are interested in rate of change, then graph rate of change.

In Figure 12, we see the population figures for a given community from 1990 to 2009. Obviously, the population is steadily increasing, but how quickly? Is the rate of population growth increasing over time, or is it decreasing? If we are interested in these questions, then simply graphing the population over time is not enough. We need to look at the rate of change directly.

Figure 12

Figure 13

The classic way to do this is by graphing the logarithm of the data. In Figure 13, we have graphed log₂ of the population over time, with the log scale printed on the right hand y-axis, and the actual population numbers printed at a log scale on the left hand axis. Now we can see that the population increased at a constant rate from 1990 to 2000, quadrupling approximately every four years, and then slowed down (to a lower constant rate) after 2000.

Graphs as a research tool

Throughout this discussion, we have considered graphs as a tool for data exploration and initial understanding. It is an iterative process — as questions arise, the data will be reprocessed and re-plotted to highlight the new issues to be examined. A good research graph must display this information directly, with a minimum of mental gymnastics, but — as with any research tool — there can be a learning curve. For example, densityplots (such as those shown in Figures 7 and 8) are in my opinion more useful than histograms for understanding how numerical data is distributed — and I am constantly surprised at the amount of explanation that they require when I show them to people who are unfamiliar with them. A number of very useful graphs that are discussed in Cleveland’s texts meet with the same reaction from people who encounter that style of graph for the first time. This is a disadvantage, relative to using a more fashionable graph, when attempting to communicate results. But the insight into the data that these graphs provide often make it worth spending the time to educate clients or peers on how to read the graph.

Even so, a good graph still may not be a quick read. As Cleveland writes:

While there is a place for rapidly-understood graphs, it is too limiting to make speed a requirement in science and technology, where the use of graphs ranges from detailed in-depth data analysis to quick presentation.
…

The important criterion for a graph is not simply how fast we can see a result; rather it is whether through the use of the graph we can see something that would have been harder to see otherwise or that could not have been seen at all.

- The Elements of Graphing Data, Chapter 2

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But researchers in other fields are not always so good about making copies of their papers easily available, and older classic papers (say, for example, Bradley Efron’s 1979 Annals of Statistics paper on the Jackknife) are often still worth reading, but are not always easy to find. Where to go?

This is a list of some resources that I’ve discovered over the years. The list isn’t comprehensive, by any means, but I offer them here because maybe you will find them helpful, too. The list, and my opinions, are biased towards research in the mathematical and computer sciences, but many of these resources are potentially useful for any research area, including the humanities.

JSTOR

JSTOR is a digital archive of over one thousand scholarly journals, covering topics in the humanities, social and physical sciences and mathematics. I love JSTOR. It is an incredibly useful resource, containing the full contents of every issue of every journal in their collection up to within 3-5 years of the present time (it’s a moving wall). The collection is full-text searchable. I use JSTOR to find classic papers in Math, Statistics, and Computer Science, as well as more recent papers that have been published in journals that are otherwise not available to me.

Access to JSTOR is available to members of participating institutions, mostly universities, but also many public libraries. I have access to JSTOR free with my San Francisco Public Library card, via the SFPL website. (I believe that any resident of California is eligible for a SFPL library card with proof of California residency; good news if you are in California and your local library doesn’t subscribe).

As a side note, San Francisco Public Library subscribes to several quite useful digital research services, including FirstSearch, the OED, Encyclopedia Brittanica, and Morningstar. Some of these other services also provide access to selected full-text articles. SFPL also participates in ILL (Interlibrary Loan) and Link+, a similar cross-library loan service. All good reasons to support your local library!

ArXiv

ArXiv is a pre-print server hosted by Cornell, serving pre-prints of papers in Physics, Mathematics, Computer Science, Quantitative Biology, Quantitative Finance and Statistics. Many important researchers use ArXiv to get around the fact that major journal publishers insist on holding the copyright to articles published in their journals. “Pre-prints” haven’t yet been published, and hence the authors are free to distribute them freely. Fields Medalist Terence Tao regularly distributes his about-to-be published work through ArXiv.

On the other hand, ArXiv has very open submission policies, so you should be more careful of the papers you find here than you would be with a refereed or curated source, such as JSTOR or PubMed Central (which we will discuss later). ArXiv has, unfortunately, more than its fair share of what Augustus de Morgan used to politely call “paradoxers“. The “Journal Reference” field of the article summaries will generally give you an indication of whether or not the paper is legitimate, in the sense of having been peer-reviewed; but note, for instance, this paper on a polynomial-time algorithm for Traveling Salesman (the Traveling Salesman problem is provably NP-complete, so a result of this magnitude would win the Clay Millennium Prize, if true).

Another side note: I’ve linked to the Amazon page on de Morgan’s Budget of Paradoxes because that was the first synopsis I found. The copyright on the book has expired, so if you are actually interested in reading it (it’s fairly funny, in places), you can find the full version on Google Books or Project Gutenberg.

CiteSeerX

CiteSeer was the original search engine and archive for online technical papers; it got me through graduate school, and my first post-PhD position at SRI. I don’t believe that the original CiteSeer system is still active, but its successor, CiteSeerX, is being developed and hosted at Penn State. It concentrates on computer science literature, as did the original. CiteSeerX builds its corpus by webcrawling, so again, the papers it finds are not necessarily refereed. Like its predecessor, CiteSeerX search results include the paper’s abstract, a BibTex citation, a list of the paper’s references, a pointer to the paper’s original location, and (usually) an archived version of the paper, in case the original link has gone dead. Good stuff.

AccessMyLibrary

AccessMyLibrary is a service that pools the periodical resources of several libraries across the United States. Any article in a periodical held by a participating library is available for free download to anyone who holds a library card in any other participating library. I find this service less useful than JSTOR: the holdings are generally newspapers and popular magazines, although there are some journals represented, as well as law and business reviews. The download format strips all of the original formatting from articles, which makes them rather ugly and a bit harder to read. I think you lose the figures, too. Still, it’s free if you have a library card, and it’s a good place to search for an article if you can’t find it anywhere else.

Questia

Questia is a for-pay service that claims to have “the world’s largest online collection of books and journal articles in the humanities and social sciences, plus magazine and newspaper articles”. Their collection is full-text searchable and, as they say, “you can read every title cover to cover”. Good luck doing so, though — articles and book chapters are not downloadable. Instead, you have to read them through Questia’s online interface, which is pretty clunky. On the plus side, they allow you to build your own “bookshelves” to collect books and articles that are relevant to you by topic or project. You can bookmark key sections, and highlight key passages. I used Questia when I was involved in research projects with psychology and organizational science aspects. I could get hold of articles or textbooks that I wanted to look at faster than through Interlibrary Loan, and more conveniently than going down to Stanford. The subscription fee at the time was cheaper than a membership to the APA or buying the articles piecemeal from Elsevier, or whoever.

Currently, Questia’s subscription fee is $19.95/month for full library access; you can also subscribe to specific collections (such as Psychology, Literature, or Philosophy) for $9.95 per collection per month.

Mendeley

Another way to find useful literature is to connect with other people out there who share your interests. Mendeley is a tool that allows you to organize your collection of research papers, share it with colleagues, and to peruse the collections of other researchers with similar interests. I haven’t used it myself; but a friend of ours who is an active and influential AI researcher recommends it. It’s certainly worth a mention.

PubMed Central

PubMed Central is a free digital archive of biomedical and life sciences journal literature, sponsored and managed by the NIH. We don’t do life science research here at Win-Vector, but I’m mentioning PubMed because of this awesome policy by the NIH:

NSF and DoD should institute similar policies, too.

Google Books, Google Scholar

Yes, they’re out there. Personally, I find them less useful than JSTOR or a subscription to (say) IEEE Explore. Google Scholar generally returns the abstracts of articles at sites that don’t provide open access to the full-text article, such as the website of the journal that published the article, or the website of a restricted research archive, like the ACM. This is useful, in that it tells you that the article exists, but it’s rather frustrating, too. I don’t find Google Scholar to be significantly more helpful than doing a general Google search on the same keywords. On the other hand, some people swear by Google Scholar, so obviously your mileage may vary.

Google Books has a very annoying habit of returning hits on your search terms, then not giving you read access to the page in question. Useless. If you happen to be doing research in an area where older books in the public domain are still of interest (for instance, my amateur interest in folklore and mythology), then Google Books can be quite helpful; of course, this situation is generally not true in technical research.

Offline: Your Local University Library

Here in the Bay Area, we are fortunate because the Stanford Library System has generous visitor access policies. The visitors’ policy statement is here; briefly, non-Stanford visitors are allowed 7 courtesy visits per year, with no borrowing privileges. For more visits, you can purchase an access card. I used the Stanford Libraries when my company was down in Mountain View, and I’m grateful for their openness. I don’t think many universities are as generous as Stanford is, but if you are near a university campus, it doesn’t hurt to check. For instance, the University of San Francisco will sell access cards to their library, with or without borrowing privileges, to non-affiliated visitors (it ain’t cheap), and allows practicing California attorneys access to their Law Library. San Francisco State has a Friends of the Library program, whereby non-affiliated visitors have access and borrowing privileges to the CSUSF library collection for $45/year.

And there you have it. Research away!

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