Biostat Methods STAT 5820/6910 Handout #9 Meta-Analysis Examples

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1 Biostat Methods STAT 5820/6910 Handout #9 Meta-Analysis Examples Example 1 A RCT was conducted to consider whether steroid therapy for expectant mothers affects death rate of premature [less than 37 weeks] newborns. Expectant mothers at risk were randomly assigned to steroid and placebo. At some set time after birth, observe numbers alive and dead. First, look at some things in SAS Dead (0) Alive (1) Steroid (1) Placebo (0) /* Read in data (line 1 from Table 12.1 of Matthews text) */ data rct; input Steroid Alive cards; ; /* Chi-square test of independence */ proc freq data=rct; tables Steroid*Alive / chisq nocol norow nopercent; weight Count; title1 'Chi-square test of independence'; run; Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Fisher's Exact Test Cell (1,1) Frequency (F) 60 Two-sided Pr <= P Sample Size =

2 /* How to quantify the difference? */ proc logistic data=rct; model Alive(event='1') = Steroid; weight Count; title1 'RCT data'; run; RCT data Probability modeled is Alive=1. Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Intercept <.0001 Steroid Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits Steroid So the odds of Alive is 73% higher in Steroid group than Placebo. 2

3 ########################################################## # Now do some things in R ########################################################## # Look at test for a single RCT rct1 <- matrix(c(36,60,496,478),ncol=2) rct1 [,1] [,2] [1,] [2,] r1 <- fisher.test(rct1) r1$p.value # Get magnitude and direction steroid <- c(1,0,1,0) alive <- c(0,0,1,1) r.wts <- c(36,60,496,478) result <- glm( alive~steroid, weight=r.wts, family='binomial') result.1 <- summary(result)$coeff round(result.1,4) Estimate Std. Error z value Pr(> z ) (Intercept) steroid exp( )

4 Example 2 The RCT in Example 1 was not the only study conducted to address the question of whether steroid therapy for expectant mothers affects death rate of premature newborns. A total of 12 RCT s (including the RCT in Example 1) were identified: By systematically combining results across these multiple studies, we can arrive at a clearer understanding of the true treatment effect. # Get results for all RCT's filepath <- " data <- read.csv(filepath) Fishers.p <- rep(na,nrow(data)) for(i in 1:nrow(data)) { mat <- matrix(as.numeric(data[i,]),ncol=2) Fishers.p[i] <- fisher.test(mat)$p.value } round(cbind(data,fishers.p),5) SteroidDead SteroidAlive PlaceboDead PlaceboAlive Fishers.p

5 # Look at Fishers p-values hist(fishers.p, main=na) Frequency Frequency Fishers.p Fishers.p # Look at Fishers composite testing approach S <- -2*sum(log(Fishers.p)) 1-pchisq(S,2*length(Fishers.p)) # Look at Stouffers composite testing approach z <- qnorm(1-fishers.p) zs <- sum(z)/sqrt(length(fishers.p)) 1-pnorm(zs) 1 ###################################### # Now look at effect size approaches ###################################### # Define effect size estimate and variance theta.hat <- log( (data[,2]+0.5)*(data[,3]+0.5) / ((data[,1]+0.5)*(data[,4]+0.5)) ) var.theta.hat <- 1/(data[,2]+0.5) + 1/(data[,3]+0.5) + 1/(data[,1]+0.5) + 1/(data[,4]+0.5) 5

6 # Look at fixed effects model w.fe <- 1/var.theta.hat theta.fe <- sum(w.fe*theta.hat) / sum(w.fe) var.fe <- 1/sum(w.FE) z.fe <- theta.fe/sqrt(var.fe) p.fe <- 2*(1-pnorm(abs(z.FE))) round(c(z.fe,p.fe),5) # Look at overall odds ratio from fixed effects model OR.FE <- exp(theta.fe) OR.FE # Now get an approximate 95% confidence interval for OR CI.FE.dn <- exp(theta.fe *sqrt(var.FE)) CI.FE.up <- exp(theta.fe *sqrt(var.FE)) round(c(ci.fe.dn,ci.fe.up),5) # automate this library(metafor) # may first need: install.packages("metafor")?rma.uni Description Function to fit the meta-analytic fixed- and random/mixed-effects models with or without moderators via linear (mixed-effects) models. See the documentation of the metaforpackage for more details on these models. Usage rma.uni(yi, vi, sei, weights, ai, bi, ci, di, n1i, n2i, x1i, x2i, t1i, t2i, m1i, m2i, sd1i, sd2i, xi, mi, ri, ti, sdi, ni, mods, measure="gen", intercept=true, data, slab, subset, add=1/2, to="only0", drop00=false, vtype="ls", method="reml", weighted=true, knha=false, level=95, digits=4, btt, tau2, verbose=false, control) 6

7 Arguments yi vi... ai bi ci di n1i n2i... mods vector of length k with the observed effect sizes or outcomes. See Details. vector of length k with the corresponding sampling variances. See Details. vector to specify the 2x2 table frequencies (upper left cell). vector to specify the 2x2 table frequencies (upper right cell). vector to specify the 2x2 table frequencies (lower left cell). vector to specify the 2x2 table frequencies (lower right cell). vector to specify the group sizes or row totals (first group/row). vector to specify the group sizes or row totals (second group/row). optional argument to include one or more moderators in the model. A single moderator can be given as a vector of length k specifying the values of the moderator. Multiple moderators are specified by giving a matrix with k rows and p' columns. Alternatively, a model formula can be used to specify the model. See Details. measure character string indicating the type of data supplied to the function. When measure="gen" (default), the observed effect sizes or outcomes and corresponding sampling variances (or standard errors) should be supplied to the function via the yi, vi, and sei arguments (only one of the two, vi or sei, needs to be specified). Alternatively, one can set measure to one of the effect size or outcome measures described under the documentation for the escalc function and specify the needed data via the appropriate arguments. data optional data frame containing the data supplied to the function. slab optional vector with unique labels for the k studies. add a non-negative number indicating the amount to add to zero cells, counts, or frequencies. See Details. to a character string indicating when the values under add should be added (either "all", "only0", "if0all", or "none"). See Details. method character string specifying whether a fixed- or a random/mixed-effects model should be fitted. A fixed-effects model (with or without moderators) is fitted when using method="fe". Random/mixed-effects models are fitted by setting method equal to one of the following: "DL", "HE", "SJ", "ML", "REML", "EB", or "HS". Default is "REML". See Details. 7

8 Measures for Dichotomous Variables In various fields (such as the health and medical sciences), the response variable measured is often dichotomous (binary), so that the data from a study comparing two different groups can be expressed in terms of a 2x2 table, such as: outcome 1 outcome 2 total group 1 ai bi n1i group 2 ci di n2i The options for the measure argument are then: "RR" for the log relative risk. "OR" for the log odds ratio. "RD" for the risk difference. "PETO" for the log odds ratio estimated with Peto's method (Yusuf et al., 1985). result <- rma.uni( ai = SteroidAlive, bi = SteroidDead, ci = PlaceboAlive, di = PlaceboDead, measure = 'OR', # report LOG odds ratio scale add = 0.5, to='all', method = 'DL', # Dersimonian-Laird slab = 1:12, # study labels data=data # dataset containing named variables ) summary(result) Random-Effects Model (k = 12; tau^2 estimator: DL) tau^2 (estimated amount of total heterogeneity): (SE = ) I^2 (total heterogeneity / total variability): 21.70% H^2 (total variability / sampling variability): 1.28 Test for Heterogeneity: Q(df = 11) = , p-val = Model Results: estimate se zval pval ci.lb ci.ub *** --- Signif. codes: 0 *** ** 0.01 *

9 # Transform LOG odds ratio estimate and CI to OR scale: exp(c(.5490,.2326,.8654)) [1] # visualize results in a forest plot forest(result) # Check for publication bias using Funnel plot funnel(result) 9

10 # Check for publication bias using Galbraith (radial) plot radial(result) # Get Galbraith test of significance: z <- result$yi/sqrt(result$vi + result$tau2) x <- 1/sqrt(result$vi + result$tau2) fit <- lm(z~x) summary(fit)$coeff Estimate Std. Error t value Pr(> t ) (Intercept) x Linear regression test of funnel plot asymmetry abline( , , lty=2) 10

11 # Check for publication bias using normal quantile plot qqnorm(result) # Include 'moderator' [covariate] -- studies 4,6,7, 8, and 12 # include only mothers with a certain complication, # while other studies excluded such mothers complication <- rep(0,nrow(data)) complication[c(4,6,7,8,12)] <- 1 # [optionally] center non-intercept columns # to preserve interpretation of intercept ctr.complication <- complication - mean(complication) result.mod <- rma.uni( ai = SteroidAlive, bi = SteroidDead, ci = PlaceboAlive, di = PlaceboDead, measure = 'OR', add = 0.5, to='all', method = 'DL', slab = 1:12, data=data, mods = ctr.complication ) summary(result.mod) Test for Residual Heterogeneity: QE(df = 10) = , p-val = Model Results: estimate se zval pval ci.lb ci.ub intrcpt < mods