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Clinical Statistical Reporting in a Multilingual World

8 Cochran Mantel Haenszel Test

The CMH procedure tests for conditional independence in partial contingency tables for a 2 x 2 x K design. However, it can be generalized to tables of X x Y x K dimensions.

8.1 Naming Convention

For the remainder of this document, we adopt the following naming convention when referring to variables of a contingency table:

  • X = exposure
  • Y = response
  • K = control

8.2 Scale

The scale of the exposure (X) and response (Y) variables dictate which test statistic is computed for the contingency table. Each test statistic is evaluated on different degrees of freedom (df):

  • General association statistic (X and Y both nominal) results in (X-1) * (Y-1) dfs
  • Row mean scores statistic (X is nominal and Y is ordinal) results in X-1 dfs
  • Nonzero correlation statistic (X and Y both ordinal) results in 1 df

8.3 Testing Strategy

8.3.1 Data

To begin investigating the differences in the SAS and R implementations of the CMH test, we decided to use the CDISC Pilot data set, which is publicly available on the PHUSE Test Data Factory repository. We applied very basic filtering conditions upfront (see below) and this data set served as the basis of the examples to follow.

# Download CDISC Pilot Dataset
# Perform filtering
library(haven)
read_xpt("https://github.com/phuse-org/TestDataFactory/raw/main/Updated/TDF_ADaM/adcibc.xpt") %>% 
  filter(EFFFL == 'Y' & ITTFL == 'Y', AVISITN == 8 & ANL01FL=='Y')

8.3.2 Schemes

In order to follow a systematic approach to testing, and to cover variations in the CMH test, we considered the traditional 2 x 2 x K design as well as scenarios where the generalized CMH test is employed (e.g. 5 x 3 x 3).

We present 5 archetype test scenarios that illustrate diverging results, possibly related to sparse data and possibly considered edge cases.

Number Schema Variables Relevant Test Description
1 2x2x2 X = TRTP, Y = SEX, K = AGEGR1 General Association TRTP and AGEGR1 were limited to two categories, overall the the groups were rather balanced
3 2x2x3 X = TRTP, Y = SEX, K = RACE General Association Gives back NaN in R because RACE is very imbalanced
6 2x5x2 X = TRTP, Y = AVAL, K = SEX Row Means Compare Row Means results for R and SAS because Y is ordinal
9 3x5x17 X = TRTP, Y = AVAL, K = SITEID Row Means SITEID has many strata and provokes sparse groups, AVAL is ordinal, therefore row means statistic applies here, R threw an error
10 5x3x3 X = AVAL, Y = AGEGR1, K = TRTP Correlation X and Y are ordinal variables and therefore the correlation statistics has to be taken here

8.4 Implementation

8.4.1 CMH in SAS

The cmh test is calculated in SAS using the PROC FREQ procedure. By default, it outputs the chi square statistic, degrees of freedom and p-value for each of the three alternative hypothesis: general association, row means differ, and nonzero correlation. It is up to the statistical analyst or statistician to know which result is appropriate for their analysis.

When the design of the contingency table is 2 x 2 x K (i.e, X == 2 levels, Y == 2 levels, K >= 2 levels), the Mantel-Haenszel Common Odds Ratio (odds ratio estimate, 95% CI, P-value) and the Breslow-Day Test for Homogeneity of the Odds Ratios (chi-square statistic, degrees of freedom, P-value) are also output.

Below is the syntax to conduct a CMH analysis in SAS:

Proc freq data = filtered_data; 
tables K * X * Y / cmh; 
* the order of K, X, and Y appearing on the line is important!;
run;

8.4.2 In R

We did not find any R package that delivers all the same measures as SAS at once. Therefore, we tried out multiple packages:

8.4.2.1 mantelhaen.test()

This is included in a base installation of R, as part of the stats package. Requires inputting data as a table or as vectors.

mantelhaen.test(x = data$x, y = data$y, z = data$k)

8.4.2.2 CMHtest()

The vcdExtra package provides results for the generalized CMH test, for each of the three model it outputs the Chi-square value and the respective p-values. Flexible data input methods available: table or formula (aggregated level data in a data frame).

library(vcdExtra)
CMHtest(Freq ~ X + Y | K , data=data, overall=TRUE)
8.4.2.2.1 Forked Version - Solution for sparse data

To tackle the issue with sparse data it is recommended that a use of solve() is replaced with MASS::ginv. This was implemented in the forked version of vcdExtra which can be installed from here:

devtools::install_github("mstackhouse/vcdExtra")

However, also the forked version for the vcdExtra package works only until a certain level of sparsity. In case of our data, it still works if the data are stratified by the pooled Site ID (SITEGR1 - 11 unique values) whereas using the unpooled Site ID (SITEID - 17 unique values) also throws an error.

8.4.2.2.2 Inconsistent Results

Exploring the vcdExtra package in detail we realized that there seems to be a bug in the implementation which we would like to highlight here.

Using an independent data example (3x2x2), we first ran the CMH test through SAS and obtained the following:

# Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
# Statistic    Alternative Hypothesis    DF       Value      Prob
# ---------------------------------------------------------------
#   1        Nonzero Correlation        1      6.4586    0.0110
#   2        Row Mean Scores Differ     2     26.0278    <.0001
#   3        General Association        2     26.0278    <.0001

When we specify the analysis in R this way, we get the following results:

CMHtest(Freq~x+y|k, data=test_eg, overall=TRUE, details=TRUE)$ALL$table
##         Chisq    Df Prob        
## cor     6.458575 1  0.01104181  
## rmeans  26.02779 2  2.229135e-06
## cmeans  6.458575 1  0.01104181  
## general 26.02779 2  2.229135e-06

By default, all 3 tests (and a 4th) are output. The results seem to match.

When we specify the analysis using an explicit type argument, we get the following results:

CMHtest(Freq~x+y|k, data=test_eg, overall=TRUE, details=TRUE, type = "ALL")$ALL$table
##         Chisq    Df Prob        
## general 26.02779 1  3.365374e-07
## rmeans  26.02779 2  2.229135e-06
## cmeans  6.458575 1  0.01104181  
## cor     6.458575 2  0.03958569

While we’ve essentially asked for everything (i.e. the default), albeit explicitly, the results do not match. The degrees of freedom seem to be mismatched with results in wrong p-values (if we assume SAS is the gold standard)

Similarly, if we specify we would only like certain tests returned, the results seem dependent on the order of specification:

# Order: cor, general, rmeans
CMHtest(Freq~x+y|k, data=test_eg, overall=TRUE, details=TRUE, types=c("cor","general","rmeans"))$ALL$table
##         Chisq    Df Prob        
## cor     6.458575 1  0.01104181  
## general 26.02779 2  2.229135e-06
## rmeans  26.02779 2  2.229135e-06
# Order: rmeans, general, cor
CMHtest(Freq~x+y|k, data=test_eg, overall=TRUE, details=TRUE, types=c("rmeans","general","cor"))$ALL$table
##         Chisq    Df Prob        
## rmeans  26.02779 1  3.365374e-07
## general 26.02779 2  2.229135e-06
## cor     6.458575 2  0.03958569

Impact: In the event of a 2x2xK design, all 3 test statistics, degrees of freedom and p-values are the same. Therefore, specifying a 2x2xK analysis any of the ways above results in correct “answer”, though they are technically mixed up.

This is why we chose as a 3x2x2, so we can clearly where things are mixed up.

We are planning to submit an issue to the vcdExtra github for the author’s review.

8.4.2.3 Epi Display package

To get the M-H common odds ratio and the homogeneity test, the epiDisplay package can be used.

library(epiDisplay) 
mhor(x,y,k, graph = FALSE)

8.5 Results

Here the results can be seen:

8.5.1 CMH Statistics

Cochran-Mantel-Haenszel Test Statistics
Schema Test Chi-Square df P-Value
R SAS R SAS R SAS
1 Correlation 0.0176704 0.017670400 1 1 0.8942487 0.89424870
Row Means 0.0176704 0.017670400 1 1 0.8942487 0.89424870
General Association 0.0176704 0.017670400 1 1 0.8942487 0.89424870
6 Correlation 0.13012 1.1472 1 1 0.71831 0.2841
Row Means 0.13012 1.1472 1 1 0.71831 0.2841
General Association 1.5771 2.5672 4 4 0.81397 0.6326
10 Correlation 0.8715295 2.738160852 1 1 0.3505322 0.09797747
Row Means 3.044527 4.407010917 4 4 0.5504018 0.35371641
General Association 5.730538 5.730538193 8 8 0.6773861 0.67738613

As it can be seen, there are two schemata where R does not provide any results:

Cochran-Mantel-Haenszel Test Statistics
Schema Test Chi-Square df P-Value
R SAS R SAS R SAS
3 Correlation NaN 0.002787130 1 1 NaN 0.95789662
Row Means NaN 2.386069847 2 2 NaN 0.30329938
General Association NaN 2.386069847 2 2 NaN 0.30329938
9 Correlation Error 0.085443119 Error 1 Error 0.77005225
Row Means Error 2.476313667 Error 2 Error 0.28991809
General Association Error 7.033878442 Error 8 Error 0.53298189

Reason for NaN in schema 3: Stratum k = AMERICAN INDIAN OR ALASKA NATIVE can not be compared because there are only values for one treatment and one gender.
Reason for Error 4:
For large sparse table (many strata) CMHTest will occasionally throw an error in solve.default(AVA) because of singularity

8.5.2 Odds Ratio

Only applicable to 2x2xK designs.

Cochran-Mantel-Haenszel Odds Ratios
Schema OR LCL UCL P-Value
R SAS R SAS R SAS R SAS
1 0.955 0.9550959 0.487 0.4865116 1.87 1.874998 0.894 NA

8.5.3 Homogenity Test

Only applicable to 2x2xK designs.

Breslow-Day Homogeneity Test of Odds Ratio
Schema Chi-Square df P-Value
R SAS R SAS R SAS
1 0.97 1.004029 1 1 0.325 0.31633757

8.6 Summary and Recommendation

Having explored the available R packages to calculate the CMH statistics, using R can only be recommended if the analysis design is equivalent to 2 x 2 x K. Then, the base mantelhaen.test() function as well as the vcdExtra package show reliable results which are equal to the output of the SAS function. The same is true for the common odds ratio even though there is a marked difference in decimals.

For the generalized version of the cmh test no R package can be recommended so far. SAS and R outputs differ substantially (possibly due to the underlying subroutines or functions) and the vcdExtra package seems to deliver inconsistent results outside the 2 x 2 x K design.