dat.begg1989.RdResults from controlled and uncontrolled studies on the effectiveness of allogeneic bone-marrow transplantation (BMT) and conventional chemotherapy (CMO) in the treatment of acute nonlymphocytic leukemia.
dat.begg1989The data frame contains the following columns:
| study | numeric | study number |
| trt | character | treatment (BMT or CMO) |
| arms | numeric | number of arms in the study (1 = uncontrolled studies; 2 = controlled studies) |
| yi | numeric | 2-year disease-free survival rates |
| sei | numeric | corresponding standard errors |
| vi | numeric | corresponding sampling variances |
The dataset includes the results from controlled and uncontrolled studies on the 2-year disease-free survival rate in patients with acute nonlymphocytic leukemia receiving either allogeneic bone-marrow transplantation (BMT) or conventional chemotherapy (CMO). In the controlled (two-arm) studies (studies 1-4), a cohort of patients in complete remission and potentially eligible for BMT was assembled, and those who consented and for whom a donor could be found received BMT, with the remaining patients used as controls (receiving CMO). In the uncontrolled (one-arm) studies (studies 5-16), only a single group was studied, receiving either BMT or CMO.
The data in this dataset were obtained from Table 1 in Begg and Pilote (1991, p. 902).
Begg, C. B., & Pilote, L. (1991). A model for incorporating historical controls into a meta-analysis. Biometrics, 47(3), 899–906. https://doi.org/10.2307/2532647
Begg, C. B., Pilote, L., & McGlave, P. B. (1989). Bone marrow transplantation versus chemotherapy in acute non-lymphocytic leukemia: A meta-analytic review. European Journal of Cancer and Clinical Oncology, 25(11), 1519–1523. https://doi.org/10.1016/0277-5379(89)90291-5
medicine, oncology, single-arm studies, multilevel models
### copy data into 'dat' and examine data
dat <- dat.begg1989
dat
#>
#> study trt arms yi sei vi
#> 1 1 BMT 2 0.4600 0.118 0.0139
#> 2 1 CMO 2 0.2500 0.074 0.0055
#> 3 2 BMT 2 0.5000 0.100 0.0100
#> 4 2 CMO 2 0.2300 0.067 0.0045
#> 5 3 BMT 2 0.4700 0.129 0.0166
#> 6 3 CMO 2 0.4200 0.086 0.0074
#> 7 4 BMT 2 0.7000 0.230 0.0529
#> 8 4 CMO 2 0.4800 0.167 0.0279
#> 9 5 BMT 1 0.4600 0.081 0.0066
#> 10 6 BMT 1 0.4300 0.034 0.0012
#> 11 7 BMT 1 0.4900 0.088 0.0077
#> 12 8 BMT 1 0.5300 0.079 0.0062
#> 13 9 CMO 1 0.2100 0.051 0.0026
#> 14 10 CMO 1 0.3200 0.039 0.0015
#> 15 11 CMO 1 0.4800 0.094 0.0088
#> 16 12 CMO 1 0.2600 0.046 0.0021
#> 17 13 CMO 1 0.3300 0.029 0.0008
#> 18 14 CMO 1 0.3800 0.033 0.0011
#> 19 15 CMO 1 0.2400 0.084 0.0071
#> 20 16 CMO 1 0.5300 0.084 0.0071
#>
### load metafor package
library(metafor)
### turn trt and arms into factors and set reference levels
dat$trt <- relevel(factor(dat$trt), ref="CMO")
dat$arms <- relevel(factor(dat$arms), ref="2")
### create data frame with the treatment differences for the controlled studies
dat2 <- data.frame(yi = dat$yi[c(1,3,5,7)] - dat$yi[c(2,4,6,8)],
vi = dat$vi[c(1,3,5,7)] + dat$vi[c(2,4,6,8)])
dat2
#> yi vi
#> 1 0.21 0.019400
#> 2 0.27 0.014489
#> 3 0.05 0.024037
#> 4 0.22 0.080789
### DerSimonian and Laird method using the treatment differences
res <- rma(yi, vi, data=dat2, method="DL", digits=2)
res
#>
#> Random-Effects Model (k = 4; tau^2 estimator: DL)
#>
#> tau^2 (estimated amount of total heterogeneity): 0 (SE = 0.02)
#> tau (square root of estimated tau^2 value): 0
#> I^2 (total heterogeneity / total variability): 0.00%
#> H^2 (total variability / sampling variability): 1.00
#>
#> Test for Heterogeneity:
#> Q(df = 3) = 1.28, p-val = 0.73
#>
#> Model Results:
#>
#> estimate se zval pval ci.lb ci.ub
#> 0.20 0.08 2.59 <.01 0.05 0.34 **
#>
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
### Begg & Pilote (1991) model incorporating the uncontrolled studies
res <- rma.mv(yi, vi, mods = ~ trt, random = ~ 1 | study,
data=dat, method="ML", digits=2)
res
#>
#> Multivariate Meta-Analysis Model (k = 20; method: ML)
#>
#> Variance Components:
#>
#> estim sqrt nlvls fixed factor
#> sigma^2 0.00 0.04 16 no study
#>
#> Test for Residual Heterogeneity:
#> QE(df = 18) = 27.65, p-val = 0.07
#>
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 16.50, p-val < .01
#>
#> Model Results:
#>
#> estimate se zval pval ci.lb ci.ub
#> intrcpt 0.32 0.02 16.47 <.01 0.28 0.36 ***
#> trtBMT 0.15 0.04 4.06 <.01 0.08 0.22 ***
#>
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
### model involving bias terms for the uncontrolled studies
res <- rma.mv(yi, vi, mods = ~ trt + trt:arms, random = ~ 1 | study,
data=dat, method="ML", digits=2)
res
#>
#> Multivariate Meta-Analysis Model (k = 20; method: ML)
#>
#> Variance Components:
#>
#> estim sqrt nlvls fixed factor
#> sigma^2 0.00 0.03 16 no study
#>
#> Test for Residual Heterogeneity:
#> QE(df = 16) = 26.72, p-val = 0.04
#>
#> Test of Moderators (coefficients 2:4):
#> QM(df = 3) = 17.64, p-val < .01
#>
#> Model Results:
#>
#> estimate se zval pval ci.lb ci.ub
#> intrcpt 0.30 0.05 6.63 <.01 0.21 0.39 ***
#> trtBMT 0.20 0.08 2.64 <.01 0.05 0.35 **
#> trtCMO:arms1 0.03 0.05 0.52 0.60 -0.07 0.12
#> trtBMT:arms1 -0.04 0.07 -0.56 0.58 -0.19 0.10
#>
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
### model with a random treatment effect
res <- rma.mv(yi, vi, mods = ~ trt, random = list(~ 1 | study, ~ trt | study),
struct="UN", tau2=c(0,NA), rho=0, data=dat, method="ML", digits=2)
res
#>
#> Multivariate Meta-Analysis Model (k = 20; method: ML)
#>
#> Variance Components:
#>
#> estim sqrt nlvls fixed factor
#> sigma^2 0.00 0.04 16 no study
#>
#> outer factor: study (nlvls = 16)
#> inner factor: trt (nlvls = 2)
#>
#> estim sqrt k.lvl fixed level
#> tau^2.1 0.00 0.00 12 yes CMO
#> tau^2.2 0.00 0.00 8 no BMT
#>
#> rho.CMO rho.BMT CMO BMT
#> CMO 1 - 4
#> BMT 0.00 1 yes -
#>
#> Test for Residual Heterogeneity:
#> QE(df = 18) = 27.65, p-val = 0.07
#>
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 16.50, p-val < .01
#>
#> Model Results:
#>
#> estimate se zval pval ci.lb ci.ub
#> intrcpt 0.32 0.02 16.47 <.01 0.28 0.36 ***
#> trtBMT 0.15 0.04 4.06 <.01 0.08 0.22 ***
#>
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
### model with a random treatment effect, but with equal variances in both arms
res <- rma.mv(yi, vi, mods = ~ trt, random = list(~ 1 | study, ~ trt | study),
struct="CS", rho=0, data=dat, method="ML", digits=2)
res
#>
#> Multivariate Meta-Analysis Model (k = 20; method: ML)
#>
#> Variance Components:
#>
#> estim sqrt nlvls fixed factor
#> sigma^2 0.00 0.00 16 no study
#>
#> outer factor: study (nlvls = 16)
#> inner factor: trt (nlvls = 2)
#>
#> estim sqrt fixed
#> tau^2 0.00 0.04 no
#> rho 0.00 yes
#>
#> Test for Residual Heterogeneity:
#> QE(df = 18) = 27.65, p-val = 0.07
#>
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 15.60, p-val < .01
#>
#> Model Results:
#>
#> estimate se zval pval ci.lb ci.ub
#> intrcpt 0.32 0.02 16.49 <.01 0.28 0.36 ***
#> trtBMT 0.15 0.04 3.95 <.01 0.07 0.22 ***
#>
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>