The functions repeatedly fit the specified model, adding one study at a time to the model.

cumul(x, ...)

# S3 method for rma.uni
cumul(x, order, digits, transf, targs, progbar=FALSE, ...)
# S3 method for rma.mh
cumul(x, order, digits, transf, targs, progbar=FALSE, ...)
# S3 method for rma.peto
cumul(x, order, digits, transf, targs, progbar=FALSE, ...)

Arguments

x

an object of class "rma.uni", "rma.mh", or "rma.peto".

order

optional argument to specify a variable based on which the studies will be ordered for the cumulative meta-analysis.

digits

optional integer to specify the number of decimal places to which the printed results should be rounded. If unspecified, the default is to take the value from the object.

transf

optional argument to specify a function to transform the model coefficients and interval bounds (e.g., transf=exp; see also transf). If unspecified, no transformation is used.

targs

optional arguments needed by the function specified under transf.

progbar

logical to specify whether a progress bar should be shown (the default is FALSE).

...

other arguments.

Details

For "rma.uni" objects, the model specified via x must be a model without moderators (i.e., either an equal- or a random-effects model).

If argument order is not specified, the studies are added according to their order in the original dataset.

When a variable is specified for order, the variable is assumed to be of the same length as the original dataset that was used in the model fitting (and if the data argument was used in the original model fit, then the variable will be searched for within this data frame first). Any subsetting and removal of studies with missing values that was applied during the model fitting is also automatically applied to the variable specified via the order argument. See ‘Examples’.

Value

An object of class c("list.rma","cumul.rma"). The object is a list containing the following components:

estimate

estimated (average) outcomes.

se

corresponding standard errors.

zval

corresponding test statistics.

pval

corresponding p-values.

ci.lb

lower bounds of the confidence intervals.

ci.ub

upper bounds of the confidence intervals.

Q

test statistics for the test of heterogeneity.

Qp

corresponding p-values.

tau2

estimated amount of heterogeneity (only for random-effects models).

I2

values of \(I^2\).

H2

values of \(H^2\).

...

other arguments.

When the model was fitted with test="t" or test="knha", then zval is called tval in the object that is returned by the function.

The object is formatted and printed with the print function. A forest plot showing the results from the cumulative meta-analysis can be obtained with forest. Alternatively, plot can also be used to visualize the results.

Note

When using the transf option, the transformation is applied to the estimated coefficients and the corresponding interval bounds. The standard errors are then set equal to NA and are omitted from the printed output.

References

Chalmers, T. C., & Lau, J. (1993). Meta-analytic stimulus for changes in clinical trials. Statistical Methods in Medical Research, 2(2), 161–172. https://doi.org/10.1177/096228029300200204

Lau, J., Schmid, C. H., & Chalmers, T. C. (1995). Cumulative meta-analysis of clinical trials builds evidence for exemplary medical care. Journal of Clinical Epidemiology, 48(1), 45–57. https://doi.org/10.1016/0895-4356(94)00106-z

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48. https://doi.org/10.18637/jss.v036.i03

See also

forest.cumul.rma for a function to draw cumulative forest plots and plot.cumul.rma for a different visualization of the cumulative results.

Examples

### calculate log risk ratios and corresponding sampling variances
dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)

### fit random-effects model
res <- rma(yi, vi, data=dat)

### cumulative meta-analysis (in the order of publication year)
cumul(res, transf=exp, order=year)
#> 
#>    estimate    zval  pvals  ci.lb  ci.ub        Q     Qp   tau2      I2      H2 
#> 1    0.4109 -1.5586 0.1191 0.1343 1.2574   0.0000 1.0000 0.0000  0.0000  1.0000 
#> 2    0.2658 -3.7967 0.0001 0.1341 0.5268   0.9315 0.3345 0.0000  0.0000  1.0000 
#> 6    0.3783 -3.9602 0.0001 0.2338 0.6120   3.1879 0.2031 0.0872 40.7090  1.6866 
#> 3    0.3675 -4.5037 0.0000 0.2377 0.5681   3.8614 0.2768 0.0763 33.9750  1.5146 
#> 10   0.3324 -5.5164 0.0000 0.2248 0.4916   7.6415 0.1056 0.0858 48.4120  1.9384 
#> 9    0.3778 -5.1875 0.0000 0.2615 0.5457  10.1854 0.0702 0.1046 60.0008  2.5000 
#> 12   0.4061 -4.7250 0.0000 0.2794 0.5901  13.2116 0.0398 0.1205 59.9982  2.4999 
#> 5    0.4545 -4.0039 0.0001 0.3089 0.6686  19.5749 0.0066 0.1780 72.3904  3.6219 
#> 7    0.4208 -4.4107 0.0000 0.2864 0.6182  22.8208 0.0036 0.2023 73.5065  3.7745 
#> 11   0.4560 -4.3588 0.0000 0.3204 0.6491  34.1203 0.0001 0.2005 81.4029  5.3772 
#> 13   0.4925 -3.9701 0.0001 0.3472 0.6987  39.6122 0.0000 0.2281 83.0110  5.8862 
#> 4    0.4517 -4.4184 0.0000 0.3175 0.6426  67.9858 0.0000 0.2732 87.0314  7.7109 
#> 8    0.4894 -3.9744 0.0001 0.3441 0.6962 152.2330 0.0000 0.3132 92.2214 12.8558 
#> 

### meta-analysis of the (log) risk ratios using the Mantel-Haenszel method
res <- rma.mh(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)

### cumulative meta-analysis
cumul(res, order=year)
#> 
#>    estimate     se     zval   pval   ci.lb   ci.ub        Q     Qp      I2      H2 
#> 1   -0.8893 0.5706  -1.5586 0.1191 -2.0077  0.2290   0.0000 1.0000  0.0000  1.0000 
#> 2   -1.3517 0.3455  -3.9124 0.0001 -2.0289 -0.6746   0.9373 0.3330  0.0000  0.9373 
#> 6   -0.8273 0.0808 -10.2371 0.0000 -0.9857 -0.6689   3.2109 0.2008 37.7127  1.6055 
#> 3   -0.8379 0.0802 -10.4490 0.0000 -0.9951 -0.6807   3.8945 0.2731 22.9687  1.2982 
#> 10  -0.8940 0.0769 -11.6254 0.0000 -1.0447 -0.7433   7.7589 0.1008 48.4460  1.9397 
#> 9   -0.8507 0.0730 -11.6480 0.0000 -0.9938 -0.7075  10.2660 0.0680 51.2956  2.0532 
#> 12  -0.8358 0.0725 -11.5275 0.0000 -0.9779 -0.6937  13.2768 0.0388 54.8084  2.2128 
#> 5   -0.7744 0.0688 -11.2623 0.0000 -0.9092 -0.6397  19.6127 0.0065 64.3088  2.8018 
#> 7   -0.7896 0.0680 -11.6188 0.0000 -0.9228 -0.6564  22.8443 0.0036 64.9803  2.8555 
#> 11  -0.6660 0.0577 -11.5373 0.0000 -0.7792 -0.5529  34.1377 0.0001 73.6362  3.7931 
#> 13  -0.6351 0.0563 -11.2811 0.0000 -0.7454 -0.5247  39.6232 0.0000 74.7622  3.9623 
#> 4   -0.7758 0.0520 -14.9267 0.0000 -0.8777 -0.6739  68.3763 0.0000 83.9126  6.2160 
#> 8   -0.4537 0.0393 -11.5338 0.0000 -0.5308 -0.3766 152.5676 0.0000 92.1346 12.7140 
#> 
cumul(res, order=year, transf=TRUE)
#> 
#>    estimate     zval   pval  ci.lb  ci.ub        Q     Qp      I2      H2 
#> 1    0.4109  -1.5586 0.1191 0.1343 1.2574   0.0000 1.0000  0.0000  1.0000 
#> 2    0.2588  -3.9124 0.0001 0.1315 0.5094   0.9373 0.3330  0.0000  0.9373 
#> 6    0.4372 -10.2371 0.0000 0.3732 0.5123   3.2109 0.2008 37.7127  1.6055 
#> 3    0.4326 -10.4490 0.0000 0.3697 0.5062   3.8945 0.2731 22.9687  1.2982 
#> 10   0.4090 -11.6254 0.0000 0.3518 0.4756   7.7589 0.1008 48.4460  1.9397 
#> 9    0.4271 -11.6480 0.0000 0.3702 0.4929  10.2660 0.0680 51.2956  2.0532 
#> 12   0.4335 -11.5275 0.0000 0.3761 0.4997  13.2768 0.0388 54.8084  2.2128 
#> 5    0.4610 -11.2623 0.0000 0.4028 0.5275  19.6127 0.0065 64.3088  2.8018 
#> 7    0.4540 -11.6188 0.0000 0.3974 0.5187  22.8443 0.0036 64.9803  2.8555 
#> 11   0.5138 -11.5373 0.0000 0.4588 0.5753  34.1377 0.0001 73.6362  3.7931 
#> 13   0.5299 -11.2811 0.0000 0.4745 0.5917  39.6232 0.0000 74.7622  3.9623 
#> 4    0.4603 -14.9267 0.0000 0.4158 0.5097  68.3763 0.0000 83.9126  6.2160 
#> 8    0.6353 -11.5338 0.0000 0.5881 0.6862 152.5676 0.0000 92.1346 12.7140 
#> 

### meta-analysis of the (log) odds ratios using Peto's method
res <- rma.peto(ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)

### cumulative meta-analysis
cumul(res, order=year)
#> 
#>    estimate     se     zval   pval   ci.lb   ci.ub        Q     Qp      I2      H2 
#> 1   -0.8604 0.5318  -1.6178 0.1057 -1.9027  0.1820   0.0000 1.0000  0.0000  1.0000 
#> 2   -1.2401 0.2912  -4.2591 0.0000 -1.8107 -0.6694   0.7279 0.3936  0.0000  0.7279 
#> 6   -0.9570 0.0897 -10.6686 0.0000 -1.1328 -0.7812   1.7723 0.4122  0.0000  0.8861 
#> 3   -0.9642 0.0885 -10.8948 0.0000 -1.1377 -0.7908   2.0147 0.5694  0.0000  0.6716 
#> 10  -1.0003 0.0823 -12.1559 0.0000 -1.1616 -0.8390   3.2425 0.5181  0.0000  0.8106 
#> 9   -0.9410 0.0776 -12.1265 0.0000 -1.0931 -0.7889   7.9341 0.1599 36.9810  1.5868 
#> 12  -0.9246 0.0771 -11.9872 0.0000 -1.0758 -0.7735  11.6730 0.0697 48.5992  1.9455 
#> 5   -0.8501 0.0730 -11.6515 0.0000 -0.9932 -0.7071  20.5359 0.0045 65.9133  2.9337 
#> 7   -0.8712 0.0724 -12.0301 0.0000 -1.0131 -0.7293  26.1159 0.0010 69.3673  3.2645 
#> 11  -0.7268 0.0615 -11.8273 0.0000 -0.8472 -0.6064  40.3188 0.0000 77.6779  4.4799 
#> 13  -0.6912 0.0599 -11.5417 0.0000 -0.8086 -0.5739  46.9968 0.0000 78.7220  4.6997 
#> 4   -0.8161 0.0531 -15.3842 0.0000 -0.9201 -0.7122  67.1837 0.0000 83.6270  6.1076 
#> 8   -0.4744 0.0407 -11.6689 0.0000 -0.5541 -0.3948 167.7302 0.0000 92.8457 13.9775 
#> 
cumul(res, order=year, transf=TRUE)
#> 
#>    estimate     zval   pval  ci.lb  ci.ub        Q     Qp      I2      H2 
#> 1    0.4230  -1.6178 0.1057 0.1492 1.1996   0.0000 1.0000  0.0000  1.0000 
#> 2    0.2894  -4.2591 0.0000 0.1635 0.5120   0.7279 0.3936  0.0000  0.7279 
#> 6    0.3840 -10.6686 0.0000 0.3221 0.4579   1.7723 0.4122  0.0000  0.8861 
#> 3    0.3813 -10.8948 0.0000 0.3206 0.4535   2.0147 0.5694  0.0000  0.6716 
#> 10   0.3678 -12.1559 0.0000 0.3130 0.4321   3.2425 0.5181  0.0000  0.8106 
#> 9    0.3902 -12.1265 0.0000 0.3352 0.4543   7.9341 0.1599 36.9810  1.5868 
#> 12   0.3967 -11.9872 0.0000 0.3410 0.4614  11.6730 0.0697 48.5992  1.9455 
#> 5    0.4274 -11.6515 0.0000 0.3704 0.4931  20.5359 0.0045 65.9133  2.9337 
#> 7    0.4184 -12.0301 0.0000 0.3631 0.4823  26.1159 0.0010 69.3673  3.2645 
#> 11   0.4835 -11.8273 0.0000 0.4286 0.5453  40.3188 0.0000 77.6779  4.4799 
#> 13   0.5010 -11.5417 0.0000 0.4455 0.5633  46.9968 0.0000 78.7220  4.6997 
#> 4    0.4421 -15.3842 0.0000 0.3985 0.4906  67.1837 0.0000 83.6270  6.1076 
#> 8    0.6222 -11.6689 0.0000 0.5746 0.6738 167.7302 0.0000 92.8457 13.9775 
#> 

### make first log risk ratio missing and fit model without study 2; then the
### variable specified via 'order' should still be of the same length as the
### original dataset; subsetting and removal of studies with missing values is
### automatically done by the cumul() function
dat$yi[1] <- NA
res <- rma(yi, vi, data=dat, subset=-2)
#> Warning: Studies with NAs omitted from model fitting.
cumul(res, transf=exp, order=year)
#> 
#>    estimate    zval  pvals  ci.lb  ci.ub        Q     Qp   tau2      I2      H2 
#> 6    0.4556 -9.4599 0.0000 0.3871 0.5362   0.0000 1.0000 0.0000  0.0000  1.0000 
#> 3    0.4514 -9.6497 0.0000 0.3841 0.5306   0.7478 0.3872 0.0000  0.0000  1.0000 
#> 10   0.3504 -4.3207 0.0000 0.2178 0.5639   4.9051 0.0861 0.1008 59.6906  2.4808 
#> 9    0.4100 -4.2685 0.0000 0.2722 0.6174   7.1487 0.0673 0.1034 66.8114  3.0131 
#> 12   0.4480 -3.6703 0.0002 0.2918 0.6878  10.0666 0.0393 0.1304 66.7246  3.0052 
#> 5    0.5075 -3.1669 0.0015 0.3335 0.7722  15.9241 0.0071 0.1704 75.5631  4.0922 
#> 7    0.4584 -3.5536 0.0004 0.2981 0.7048  19.3442 0.0036 0.2132 77.6086  4.4660 
#> 11   0.4970 -3.6803 0.0002 0.3425 0.7212  29.4342 0.0001 0.1880 83.2856  5.9829 
#> 13   0.5390 -3.3443 0.0008 0.3752 0.7743  34.5952 0.0000 0.2077 84.0877  6.2845 
#> 4    0.4835 -3.7288 0.0002 0.3300 0.7084  64.2052 0.0000 0.2831 89.1076  9.1808 
#> 8    0.5262 -3.3578 0.0008 0.3618 0.7655 144.6390 0.0000 0.3139 93.3044 14.9352 
#>