Meta-analysis for a diagnostic (no date)

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There is no real consensus yet on how to best combine data from several studies of a diagnostic test. I will outline a few approaches that seem to make sense. In addition to this page, I have a general overview on meta-analysis and a non-technical introduction on the practical interpretation of a meta-analysis.

Direct analysis of sensitivity/specificity

The simplest overall estimate of sensitivity (sens) or specificity (spec) is to just combine all the studies in a pot and stir. Just count the number of true positives (tp), false negatives (fn), true negatives (tn) and false positives (fp) in each study. The overall sensitivity would have the sum of the individual true positive values in the numerator and the sum of the individual true positive plus false negative values in the denominator.

This is equivalent to a weighted average of the individual sensitivities where the weights for each individual study is simply the individual true positive plus false negative values. You would calculate an overall estimate of sp.

The tricky part comes when you try to define a confidence interval for the overall estimate. This confidence interval is effectively a combination of the standard errors that you would assign to each individual study.

A first attempt might be to define the standard error of an individual study using the classic binomial formula. Writing the standard error in terms of true positive and false negative values, you would get

The problem with this formula for the standard error is that it gives less weight to studies where sensitivity is close to 50% and greater weight to studies where sensitivity is much smaller than 50% or much larger than 50%. Another problem occurs when one or more of the sensitivities is 100%. The standard error using a binomial distribution equals zero for those studies with 100% sensitivity, which seems at first like a good thing. But when one study has  standard error of zero, the meta-analysis model will try to give it an infinite weight, which is not at all a good thing.

One way to avoid some of these problems is to estimate the standard error, not using the individual sensitivities, but the overall sensitivity.

Since the numerator is now the same for every study, you no longer have the problem where studies with sensitivities near 50% get much smaller weights than studies with sensitivities much smaller or much larger than 50%. This approach also avoids the problem when a study has 100% sensitivity.

It's interesting to note that, the overall estimate and the standard error for the overall sensitivity using this particular meta-analysis model with a fixed effects estimate matches perfectly with the traditional binomial confidence interval that you might apply. This is easy enough to show because

which implies that

For a random effects model, the results are a little more complicated and they do not exactly match the traditional binomial confidence interval formula.

Example: In an article describing systematic reviews of diagnostic and screening tests,

data from 20 studies of endovaginal ultrasonography for detecting endometrial cancer are presented. I typed the data in as a comma separated file.

Abu Hmeidan,81,5,186,273
Karlsson (a),112,2,414,601
Karlsson (b),14,1,33,57
Nasri (a),7,0,14,38
Nasri (b),6,0,24,59

and here is the R code to read in an compute the meta-analysis models.

f0 <- "X:/webdata/EndovaginalUltrasonography.csv"
deeks.example.dat <- read.csv(f0)
sens <- tp / (tp + fn)
sens.overall <- sum(tp) / sum(tp + fn)
spec <- tn / (tn + fp)
spec.overall <- sum(tn) / sum(tn + fp)


The last three lines create a graph of the data, which is shown below. The par() function adjusts the margins of the graph to make more effective use of the available space on the screen. The plot() function creates the axes and draws a circle at each individual sens, 1-spec pair. The points() command adds a big plus sign at the overall estimate.

Plotting 1-spec on the x-axis, which seems odd, but it is intended to have the same orientation as an ROC curve. In fact, this plot is often called an SROC (Summary Receiver Operating Characteristic) plot.

I experimented with trying to show the confidence limits for each study in the graph itself, by drawing rectangles with the width representing confidence limits for 1-spec and the height representing confidence limits for sens. Unfortunately, this graph was too cluttered to be useful.

The computations for the actual meta-analysis are shown below. The code is a bit cryptic perhaps, but I am using "te" as an abbreviation for "treatment effect" and "se" as an abbreviation for "standard error." The metagen() function has similar notation. The only thing that is a bit confusing perhaps is the sm= portion. The letters "sm" stand for "summary measure. This is a label that metagen uses to make the output look nicer.

te1 <- sens
se1 <- sqrt(sens.overall * (1 - sens.overall) / (tp + fn)) <- metagen(TE=te1, seTE=se1, studlab=study, sm="Sensitivity")
te2 <- spec
se2 <- sqrt(spec.overall * (1 - spec.overall) / (tn + fp)) <- metagen(TE=te2, seTE=se2, studlab=study, sm="Specificity")

and here is the output


        Sensitivity           95%-CI %W(fixed) %W(random)
Abu Hmeidan  0.9419 [0.8997; 0.9840]     18.82      10.27
Auslender    1.0000 [0.9022; 1.0978]      3.50       5.62
Botsis       1.0000 [0.8617; 1.1383]      1.75       3.61
Cacclatore   1.0000 [0.8044; 1.1956]      0.88       2.10
Chan         0.8824 [0.7875; 0.9772]      3.72       5.81
Dorum        0.8000 [0.6990; 0.9010]      3.28       5.42
Goldstein    1.0000 [0.6088; 1.3912]      0.22       0.60
Granberg     1.0000 [0.9078; 1.0922]      3.94       5.99
Hanggi       0.8571 [0.7718; 0.9425]      4.60       6.47
Karlsson (a) 0.9825 [0.9458; 1.0191]     24.95      10.77
Karlsson (b) 0.9333 [0.8323; 1.0344]      3.28       5.42
Klug         0.8750 [0.7367; 1.0133]      1.75       3.61
Malinova     1.0000 [0.9482; 1.0518]     12.47       9.37
Nasri (a)    1.0000 [0.8521; 1.1479]      1.53       3.27
Nasri (b)    1.0000 [0.8403; 1.1597]      1.31       2.91
Petrl        0.9474 [0.8576; 1.0371]      4.16       6.16
Taviani      1.0000 [0.7233; 1.2767]      0.44       1.15
Varner       0.5000 [0.2233; 0.7767]      0.44       1.15
Weigel       1.0000 [0.9357; 1.0643]      8.10       8.21
Wolman       1.0000 [0.8044; 1.1956]      0.88       2.10

Number of trials combined: 20

                Sensitivity           95%-CI        z  p.value
Fixed effects model  0.9584 [0.9401; 0.9767] 102.6404 < 0.0001
Random effects model 0.9481 [0.9171; 0.9792]  59.8249 < 0.0001

Quantifying heterogeneity:
tau^2 = 0.002; H = 1.43 [1.1; 1.85]; I^2 = 51% [18.1%; 70.7%]

Test of heterogeneity:
    Q d.f. p.value
38.75  19   0.0048

Method: Inverse variance method


        Specificity           95%-CI %W(fixed) %W(random)
Abu Hmeidan  0.4052 [0.3606; 0.4498]     15.27       5.83
Auslender    0.3478 [0.2665; 0.4292]      4.59       5.46
Botsis       0.1250 [0.0347; 0.2153]      3.73       5.35
Cacclatore   0.7317 [0.5825; 0.8810]      1.36       4.49
Chan         0.3000 [0.1648; 0.4352]      1.66       4.71
Dorum        0.4000 [0.2963; 0.5037]      2.83       5.17
Goldstein    0.5926 [0.4087; 0.7765]      0.90       3.97
Granberg     0.2038 [0.1275; 0.2801]      5.22       5.52
Hanggi       0.1912 [0.0753; 0.3071]      2.26       4.99
Karlsson (a) 0.4079 [0.3779; 0.4379]     33.78       5.93
Karlsson (b) 0.3667 [0.2659; 0.4674]      3.00       5.21
Klug         0.2573 [0.1842; 0.3304]      5.69       5.56
Malinova     0.4262 [0.3039; 0.5486]      2.03       4.90
Nasri (a)    0.2692 [0.1367; 0.4018]      1.73       4.75
Nasri (b)    0.2892 [0.1843; 0.3941]      2.76       5.15
Petrl        0.7328 [0.6493; 0.8163]      4.36       5.43
Taviani      0.4615 [0.3085; 0.6146]      1.30       4.43
Varner       0.3077 [0.0426; 0.5728]      0.43       2.91
Weigel       0.5583 [0.4834; 0.6331]      5.42       5.54
Wolman       0.3600 [0.2248; 0.4952]      1.66       4.71

Number of trials combined: 20

               Specificity            95%-CI       z  p.value
Fixed effects model  0.3894 [0.3719; 0.4068] 43.7721 < 0.0001
Random effects model 0.3845 [0.3216; 0.4475] 11.9685 < 0.0001

Quantifying heterogeneity:
tau^2 = 0.0172; H = 3.26 [2.77; 3.85]; I^2 = 90.6% [86.9%; 93.2%]

Test of heterogeneity:
     Q d.f.  p.value
202.17   19 < 0.0001

Method: Inverse variance method

Notice that there is substantial evidence of heterogeneity in both the sensitivity and specificity values.

Analysis of sensitivity/specificity on the log odds scale

Another approach is to transform the sensitivity/specificity to the log odds scale before entering the data into a meta-analysis model. The log odds transformation is a common transformation for binomial data and serves as the heart of a logistic regression model. The standard error for the log odds sensitivity has a nice simple approximation. To derive this, you have to remember a simple formula about variances of a function.

This formula relies on two things you forgot from your days of calculus, how to take a derivative and how to apply a Taylor series expansion.

The details are tedious, but not difficult. When you use this formula on the log odds function, you get the following approximation.

Compare this to the standard error for sensitivity shown above. The numerator for the standard error has now moved in with its downstairs neighbor, leaving the upstairs empty. For the log odds for sensitivity, this the opposite problem from the sensitivity. Studies with sensitivity close to 50% have greater weight on the log odds scale than studies with sensitivity larger than 50%.

You can simplify this formula further. Note that the denominator of sensi can cancel out the tpi+fni term right next to it. With a bit more algebra, you can get

The log odds transformation also has some problems when the sensitivity is 100%. A simple fix is to add an arbitrary constant (usually 0.5) to both the numerator and denominator. Another approach would be to use the more complex formula listed above, but substitute the overall sensitivity for the individual sensitivity.

Example: Let's use the example in Deeks 2001 again. Here is the R code to compute log odds and analyze the data in a meta-analysis model. Note that the pmax function replaces the zeros in fn with 0.5.

logit <- function(p) {log(p)-log(1-p)}
fn.adj <- pmax(fn,0.5)
sens <- tp/(tp+fn.adj)
te3 <- logit(sens)
se3 <- sqrt(1/tp+1/fn.adj) <- metagen(TE=te3,seTE=se3,studlab=study,sm="Log Odds Sens")
spec <- tn/(tn+fp)
te4 <- logit(spec)
se4 <- sqrt(1/tn+1/fp) <- metagen(TE=te4,seTE=se4,studlab=study,sm="Log Odds Spec")

Here is the output. Using the summary function results in a briefer output because the results of individual studies are not shown.


Number of trials combined: 20

              Log Odds Sens           95%-CI       z  p.value
Fixed effects model  2.4775 [2.0562; 2.8987] 11.5269 < 0.0001
Random effects model 2.4761 [2.0318; 2.9204] 10.9228 < 0.0001

Quantifying heterogeneity:
tau^2 = 0.0551; H = 1.03 [1; 1.27]; I^2 = 5.4% [0%; 38.1%]

Test of heterogeneity:
    Q d.f. p.value
20.07   19  0.3901

Method: Inverse variance method


Number of trials combined: 20

               Log Odds Spec             95%-CI        z  p.value
Fixed effects model  -0.4277 [-0.5036; -0.3518] -11.0403 < 0.0001
Random effects model -0.5033 [-0.7668; -0.2399] -3.7446    0.0002

Quantifying heterogeneity:
tau^2 = 0.292; H = 3.07 [2.58; 3.64]; I^2 = 89.4% [85%; 92.5%]

Test of heterogeneity:
     Q d.f.  p.value
178.76   19 < 0.0001

Method: Inverse variance method

You need to do a few additional calculations to get sensitivity transformed back to the original measurement scale. You can define a function in R to do this calculation for you. I call it the expit function, which is the inverse of the logit function.

expit <- function(log.odds) {exp(log.odds)/(1+exp(log.odds))}

With this function, you can now take the estimates and confidence limits on the log odds scale and transform them back to the original scale.

attach( <- TE.fixed+c(0,-1.96,1.96)*seTE.fixed

92.3 88.7 94.8 <- TE.random+c(0,-1.96,1.96)*seTE.random

92.2 88.4 94.9

attach( <- TE.fixed+c(0,-1.96,1.96)*seTE.fixed

39.5 37.7 41.3 <- TE.random+c(0,-1.96,1.96)*seTE.random

37.7 31.7 44.0

The estimated sensitivity and 95% confidence limits under the fixed effects model are 92.3% (88.7% to 94.8%). The estimates and limits change only slightly under than random effects model. The estimated specificity and 95% confidence limits under the fixed effect model are 39.5% (37.7% to 41.3%). Under the random effects model, the estimate is a bit lower and the confidence limits are much wider.

Analysis of the diagnostic odds ratio

A third approach is to compute the diagnostic odds ratio, which compares the odds for sensitivity to the odds for specificity.

Notice how the denominator looks like we accidentally switched things. That was not a mistake. The diagnostic odds ratio is effectively the odds of TPR (the true positive rate or sens) divided by the odds of FPR (the false positive rate or 1-spec).

The first advantage of this approach is that you can use well-known approaches for combining multiple odds ratios. The other advantage is that is analyzes sensitivity and specificity as a pair. Some studies may exhibit heterogeneity in the individual sensitivity or specificity values because one researcher may have tried to maximize sensitivity at the expense of specificity, another may have tried to maximize specificity at the expense of sensitivity, and a third may have tried to balance the two. If there is heterogeneity, then the overall estimates of sensitivity and specificity may be too low.

Although there are no guarantees, the diagnostic odds ratio should exhibit less heterogeneity. The problem with the diagnostic odds ratio is that no one has a very good feel on what it actually represents. One way of thinking about the diagnostic odds ratio is to swap a couple of terms in the fraction.

So you might interpret the diagnostic odds ratio as the spread between the two likelihood ratios. If, for example, the likelihood ratio for a positive test is 10 and is 0.5 for a negative test, then there is a 20 fold change. Another way of interpreting this is that the post-test odds would be 20 fold higher for a positive test than for a negative test.

The book on meta-analysis by Sutton et al suggests that  you model the heterogeneity in the diagnostic odds ratio using the following regression model

You might recognize D as the diagnostic odds ratio. The variable S is a bit harder to visualize, but you can rewrite it as

This represents the tendency of an individual study to skew the test more towards sensitivity or more towards specificity.

Here's an example of the problems that can happen when different studies skew more towards sensitivity and others more towards specificity. Imagine a diagnostic test that takes on a range of values. This test follows a bell shaped curve both in the diseased and the healthy populations and the two bell curves are set two standard deviations apart. You could set a cutpoint to maximize specificity, to maximize sensitivity, or something in between.

This series of graphs shows what happens across a range of cutpoints.

When you graph the data on an SROC plot, you get a nice distribution of values. Notice, however, that the average of all these sensitivities and specificities is pushed further away from the upper left hand corner than any of the individual sensitivity/specificity pairs.

By fitting a model to the diagnostic odds ratio, and assessing heterogeneity in that odds ratio, you hope to avoid this obvious underestimate of sensitivity and specificity.

When you fit the regression model, you are hoping is that the slope term is zero. That tells you that the estimated intercept is a valid estimate across the range of S values.

It's unclear whether to use a weighted regression model or an unweighted regression model for these data.

fn.adj <- pmax(fn,0.5)
tpr <- tp/(tp+fn.adj)
fpr <- fp/(tn+fp)
d <- logit(tpr)-logit(fpr)
s <- logit(tpr)+logit(fpr)
se.d <- sqrt(1/tp+1/fn.adj+1/tn+1/fp)
w <- 1/se.d^2
unweighted.regression <- lm(d~s)
weighted.regression <- lm(d~s,weights=w)

For this data set, it appears that there is a non-zero slope, which makes interpretation of the combined diagnostic odds ratio problematic. <- metagen(TE=d,seTE=se.d,studlab=study,sm="Log Diagnostic Odds Ratio")

Number of trials combined: 20

Log Diagnostic Odds  Ratio   95%-CI          z      p.value
Fixed effects model  1.9772 [1.5400; 2.4145] 8.8633 < 0.0001
Random effects model 1.9732 [1.3618; 2.5847] 6.3249 < 0.0001

Quantifying heterogeneity:
tau^2 = 0.6555; H = 1.27 [1; 1.67]; I^2 = 38.4% [0%; 64%]

Test of heterogeneity:
    Q d.f. p.value
30.87  19  0.0418

Method: Inverse variance method

Additional reading