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analyses:vanhouwelingen1993 [2021/10/22 14:45] – Wolfgang Viechtbaueranalyses:vanhouwelingen1993 [2022/08/03 11:22] (current) – Wolfgang Viechtbauer
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 ==== The Methods and Data ==== ==== The Methods and Data ====
  
-The paper by van Houwelingen et al. (1993) is an early (and unfortunately often overlooked) paper on meta-analytic methods for 2×2 table data that describes a variety of rather sophisticated models and methods, including the fixed/random-effects conditional logistic model, a nonparametric mixture model based on Laird (1978), and the bivariate binomial-normal model. The models and methods are illustrated with data from 27 studies examining the effectiveness of histamine H2 antagonists (cimetidine or ranitidine) in treating patients with acute upper gastrointestinal hemorrhage. The dataset can be loaded with:+The paper by van Houwelingen et al. (1993) is an early (and unfortunately often overlooked) paper on meta-analytic methods for 2×2 table data that describes a variety of rather sophisticated models and methods, including the equal- and random-effects conditional logistic models, a nonparametric mixture model based on Laird (1978), and the bivariate binomial-normal model. The models and methods are illustrated with data from 27 studies examining the effectiveness of histamine H2 antagonists (cimetidine or ranitidine) in treating patients with acute upper gastrointestinal hemorrhage. The dataset can be loaded with:
 <code rsplus> <code rsplus>
 library(metafor) library(metafor)
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 No information about this outcome was available for 2 trials, so the results below are based on the remaining 25 trials. Also, note that van Houwelingen et al. (1993) analyze the log odds ratios of persistent or recurrent bleedings in the control group versus the treatment group, so that positive values indicate a lower risk in the treatment group and hence a positive treatment effect. No information about this outcome was available for 2 trials, so the results below are based on the remaining 25 trials. Also, note that van Houwelingen et al. (1993) analyze the log odds ratios of persistent or recurrent bleedings in the control group versus the treatment group, so that positive values indicate a lower risk in the treatment group and hence a positive treatment effect.
  
-==== Fixed-Effects Conditional Logistic Model ====+==== Equal-Effects Conditional Logistic Model ====
  
-Among the models considered by the authors is the fixed-effects conditional logistic model (described as a "likelihood based Mantel-Haenszel-type procedure" in the article). The model results from conditioning on the total number of cases within each study, leading to the non-central hypergeometric distribution for the 2×2 tables. Figure 2 in the paper (p. 2277) shows the likelihoods of the log odds ratios based on the non-central hypergeometric distributions for the individual studies. An analogous figure can be produced with:+Among the models considered by the authors is the equal-effects conditional logistic model (described as a "likelihood based Mantel-Haenszel-type procedure" in the article). The model results from conditioning on the total number of cases within each study, leading to the non-central hypergeometric distribution for the 2×2 tables. Figure 2 in the paper (p. 2277) shows the likelihoods of the log odds ratios based on the non-central hypergeometric distributions for the individual studies. An analogous figure can be produced with:
 <code rsplus> <code rsplus>
 llplot(measure="OR", ai=b.xci, n1i=nci, ci=b.xti, n2i=nti, data=dat, llplot(measure="OR", ai=b.xci, n1i=nci, ci=b.xti, n2i=nti, data=dat,
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 One study (number 10) has a single zero cell (none of the patients in the treatment group experienced persistent or recurrent bleedings), which implies that the MLE of the odds ratio is infinite. The dashed line in the figure corresponds to this study. Also, one study (number 14) has two zero cells (none of the patients in the entire study experienced persistent or recurrent bleedings), which implies a flat likelihood for the odds ratio. The dotted line in the figure corresponds to this study.((Note that the ''llplot()'' function would have omitted this study from the plot by default. With ''drop00=FALSE'', we can also see the (flat) likelihood for this study.)) However, for most studies, the likelihood is concentrated within the range $-1$ to $1$. Hence, the MLE of the log odds ratio should also fall somewhere within this range. One study (number 10) has a single zero cell (none of the patients in the treatment group experienced persistent or recurrent bleedings), which implies that the MLE of the odds ratio is infinite. The dashed line in the figure corresponds to this study. Also, one study (number 14) has two zero cells (none of the patients in the entire study experienced persistent or recurrent bleedings), which implies a flat likelihood for the odds ratio. The dotted line in the figure corresponds to this study.((Note that the ''llplot()'' function would have omitted this study from the plot by default. With ''drop00=FALSE'', we can also see the (flat) likelihood for this study.)) However, for most studies, the likelihood is concentrated within the range $-1$ to $1$. Hence, the MLE of the log odds ratio should also fall somewhere within this range.
  
-The results from a corresponding fixed-effects model can be obtained with:+The results from a corresponding equal-effects model can be obtained with:
 <code rsplus> <code rsplus>
 res <- rma.glmm(measure="OR", ai=b.xci, n1i=nci, ci=b.xti, n2i=nti, res <- rma.glmm(measure="OR", ai=b.xci, n1i=nci, ci=b.xti, n2i=nti,
-                data=dat, model="CM.EL", method="FE")+                data=dat, model="CM.EL", method="EE")
 summary(res, digits=2) summary(res, digits=2)
 </code> </code>
 <code output> <code output>
-Fixed-Effects Model (k = 24)+Equal-Effects Model (k = 24)
 Model Type: Conditional Model with Exact Likelihood Model Type: Conditional Model with Exact Likelihood
  
-  logLik  deviance       AIC       BIC      AICc   +  logLik  deviance       AIC       BIC      AICc 
-  -53.68     40.34    109.36    110.54    109.54  +  -53.68     40.34    109.36    110.54    109.54
  
-Tests for Heterogeneity:  +Tests for Heterogeneity: 
-Wld(df = 23) = 32.54, p-val = 0.09+Wld(df = 23) = 32.52, p-val = 0.09
 LRT(df = 23) = 40.34, p-val = 0.01 LRT(df = 23) = 40.34, p-val = 0.01
  
 Model Results: Model Results:
  
-estimate       se     zval     pval    ci.lb    ci.ub           +estimate    se  zval  pval  ci.lb  ci.ub 
-    0.12     0.10     1.22     0.22    -0.07     0.32          +    0.12  0.10  1.22  0.22  -0.07   0.32
  
 --- ---
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
 </code> </code>
 These results match what is reported on page 2275. In particular, the log odds ratio is estimated to be $\hat{\theta} = 0.12$ (with 95% confidence interval: $-0.07$ to $0.32$). The log likelihood for this model is $ll = -53.68$. These results match what is reported on page 2275. In particular, the log odds ratio is estimated to be $\hat{\theta} = 0.12$ (with 95% confidence interval: $-0.07$ to $0.32$). The log likelihood for this model is $ll = -53.68$.
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 </code> </code>
 <code output> <code output>
-Fixed-Effects Model (k = 25)+Equal-Effects Model (k = 25)
  
-Test for Heterogeneity: +I^2 (total heterogeneity / total variability):  33.94% 
 +H^2 (total variability / sampling variability): 1.51 
 + 
 +Test for Heterogeneity:
 Q(df = 23) = 34.82, p-val = 0.05 Q(df = 23) = 34.82, p-val = 0.05
  
 Model Results (log scale): Model Results (log scale):
  
-estimate       se     zval     pval    ci.lb    ci.ub  +estimate    se  zval  pval  ci.lb  ci.ub 
-    0.12     0.10     1.22     0.22    -0.07     0.32 +    0.12  0.10  1.22  0.22  -0.07   0.32
  
 Model Results (OR scale): Model Results (OR scale):
  
-estimate    ci.lb    ci.ub  +estimate  ci.lb  ci.ub 
-    1.13     0.93     1.37 +    1.13   0.93   1.37
  
 Cochran-Mantel-Haenszel Test:    CMH =  1.37, df = 1,  p-val = 0.24 Cochran-Mantel-Haenszel Test:    CMH =  1.37, df = 1,  p-val = 0.24
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 Model Type: Conditional Model with Exact Likelihood Model Type: Conditional Model with Exact Likelihood
  
-  logLik  deviance       AIC       BIC      AICc   +  logLik  deviance       AIC       BIC      AICc 
-  -52.99     38.96    109.98    112.33    110.55  +  -52.99     38.96    109.98    112.34    110.55
  
 tau^2 (estimated amount of total heterogeneity): 0.12 (SE = 0.14) tau^2 (estimated amount of total heterogeneity): 0.12 (SE = 0.14)
 tau (square root of estimated tau^2 value):      0.35 tau (square root of estimated tau^2 value):      0.35
-I^2 (total heterogeneity / total variability):   30.90%+I^2 (total heterogeneity / total variability):   30.86%
 H^2 (total variability / sampling variability):  1.45 H^2 (total variability / sampling variability):  1.45
  
-Tests for Heterogeneity:  +Tests for Heterogeneity: 
-Wld(df = 23) = 32.54, p-val = 0.09+Wld(df = 23) = 32.52, p-val = 0.09
 LRT(df = 23) = 40.34, p-val = 0.01 LRT(df = 23) = 40.34, p-val = 0.01
  
 Model Results: Model Results:
  
-estimate       se     zval     pval    ci.lb    ci.ub           +estimate    se  zval  pval  ci.lb  ci.ub 
-    0.17     0.14     1.28     0.20    -0.09     0.44          +    0.17  0.14  1.28  0.20  -0.09   0.44
  
 --- ---
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
 </code> </code>
-These results match what is reported on page 2277. In particular, the estimated average log odds ratio is $\hat{\mu} = 0.17$ (with 95% confidence interval: $-0.09$ to $0.44$).((Note that this interval is slightly different than the confidence profile interval reported by the authors on page 2278. The interval obtained here is a Wald-type interval.)) The estimated amount of heterogeneity is $\hat{\tau}^2 = .35^2$. The log likelihood for this model is $ll = -52.99$.+These results match what is reported on page 2277. In particular, the estimated average log odds ratio is $\hat{\mu} = 0.17$ (with 95% confidence interval: $-0.09$ to $0.44$),((Note that this interval is slightly different than the confidence profile interval reported by the authors on page 2278. The interval obtained here is a Wald-type interval.)) while the estimated amount of heterogeneity is $\hat{\tau}^2 = .35^2$. The log likelihood for this model is $ll = -52.99$.
  
 ==== Mixture Model ==== ==== Mixture Model ====
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 </code> </code>
 <code output> <code output>
-Computer Assisted Mixture Analysis:  +Computer Assisted Mixture Analysis: 
-  + 
-Data consists of 25 observations (rows).  +Data consists of 25 observations (rows). 
-The Mixture Analysis identified 2 components of a gaussian distribution:  +The Mixture Analysis identified 2 components of a gaussian distribution: 
- +
 DETAILS: DETAILS:
           p         mean           p         mean
 1 0.8183766 -0.008554017 1 0.8183766 -0.008554017
 2 0.1816234  1.036847813 2 0.1816234  1.036847813
- +
 Log-Likelihood: -28.87813     BIC: 67.41289 Log-Likelihood: -28.87813     BIC: 67.41289
 </code> </code>
analyses/vanhouwelingen1993.txt · Last modified: 2022/08/03 11:22 by Wolfgang Viechtbauer