CMHTEST procedure

Performs the Cochran-Mantel-Haenszel test (D.A. Murray).

Options

`PRINT` = string token	Controls printed output (`test`); default `test`
`CLASSIFICATION` = factors	Classifying factors for a `DATA` variate or classifying factors for the R×C tables in a `DATA` table
`CONTINUITY` = string token	Continuity correction for 2×2×K Mantel-Haenszel test (`correct`, `none`); default `corr`
`CIPROBABILITY` = scalar	Size of confidence interval for common odds ratio in 2×2×K tables; default 0.95

Parameters

`DATA` = tables or variates	Data values
`STATISTIC` = scalars	Save the test statistic
`PROBABILITY` = scalars	Save the probability for the test
`ODDSRATIO` = scalars	Save the common odds ratio for the 2×2×K table case
`LOWER` = scalars	Save lower limit of the confidence interval of odds ratio
`UPPER` = scalars	Save upper limit of the confidence interval of odds ratio

Description

CMHTEST performs the Cochran-Mantel-Haenszel test for average partial association between two nominal variables adjusting for control variables. The data are represented by a series of K (R×C) contingency tables, where K represents the strata for the control variables. If there are two or more control variables then these are combined to form a single factor (K) with a level for every combination of the control factors. For the case where there are two dichotomous variables of interest, i.e. a series of K (2×2) tables, CMHTEST calculates the Mantel-Haenszel chi-square statistic, and an overall estimate of relative risk as described in Mantel & Haenszel (1959). Otherwise the Generalized Cochran-Mantel-Haenszel test is used, as in Landis et al. (1978).

The data can be supplied as a table using the DATA parameter where the first two classifying factors of the table indicate the variables of interest, and the remaining factors are combined to form a factor with a level for every combination of the remaining factors. If the first two classifying factors are not the ones of interest, then the CLASSIFICATION option can be used to supply the names of the classifying factors to use. The data can also be supplied in variates, with the CLASSIFICATION option set to the classifying factors and the first two factors in the list indicating the variables of interest. For a series of K (2×2) tables the CONTINUITY option can be used to control whether to apply a continuity correction to the Mantel-Haenszel chi-square test.

The PRINT option controls printed output, with settings:

`test`	the test statistic and probability, also the common odds ratio and confidence interval when there are K (2×2) tables

A 95% confidence interval is calculated for the common odds ratio, but this can be changed by setting the CIPROBABILITY option to the required value (between 0 and 1).

The test statistic can be saved using the STATISTIC parameter, and the probability can be saved using the PROBABILITY parameter. For a series of K (2×2) tables the odds ratio, lower and upper odds-ratio confidence interval can be saved with the ODDSRATIO, LOWER and UPPER parameters respectively.

Options: PRINT, CLASSIFICATION, CONTINUITY, CIPROBABILITY.

Parameters: DATA, STATISTIC, PROBABILITY, ODDSRATIO, LOWER, UPPER.

Method

For each table i, i = 1…K

a_i	b_i	n_1i
c_i	d_i	n_2i
m_1i	m_2i	N_i

the Mantel-Haenszel Test is calculated by:

MH = ( |( ∑ a_i – ∑((n_1i × m_1i) / N_i) )| – 0.5 )²

/ ∑( (n1i × n_2i × m_1i × m_2i) / (N_i² × (N_i– 1)) )

where the continuity correction (0.5) is used if option CONTINUITY=correct. The common odds-ratio is calculated by

OR = ∑_{i=1 to K}R_i / ∑_{i=1 to K}S_i

where

R_i = (a_i × d_i) / N_i

S_i = (b_i × c_i) / N_i

The variance for the odds-ratio is estimated using the method outlined in Robins et al. (1986).

The Generalized Cochran-Mantel-Haenszel test is calculated using the method of Landis et al. (1978).

Action with `RESTRICT`

If a parameter is restricted the statistics will be calculated using only those units included in the restriction.

References

Landis J,L., Heyman, E,R. & Koch, G.G. (1978). Average Partial Association in Three-way Contingency Tables: a Review and Discussion of Alternative Tests. International Statistical Review, 46, 237-254.

Mantel N. & Haenszel W. (1959). Statistical Aspects of the Analysis of Data From Retrospective Studies of Disease. Journal National Cancer Institute, 22(4), 719-748.

Robins J, Breslow N, & Greenland S. (1986). Estimators of the Mantel-Haenszel variance consistent in both sparse data and large-strata limiting models. Biometrics, 42, 311-323.

Example

CAPTION 'CMHTEST example 1',\
        !t('Data from Mantel & Haenszel (1959) Study of women',\
        'with epidermoid and undifferentiated pulmonary carcinoma');\
        STYLE=meta,plain
FACTOR  [LEVELS=2; LABELS=!t('Pulmonary carinoma','Controls')] cases
FACTOR  [LEVELS=2; LABELS=!t('Smoker','Nonsmoker')] Smoke
FACTOR  [LEVELS=4; LABELS=!t('under 45','45-54','55-64','over 65')] Age
FACTOR  [LEVELS=3; LABELS=!t('Housewives','White-collar','Other')] Occupation
TABLE   [CLASS=cases,Smoke,Age,Occupation; MARGINS=no] pulmonary
READ    pulmonary
0 3 1 2 2 4 3 2 0 0 0 1 2 0 0 5 2 1 6 4 6 11 6 3 0 2 3 1 2 1 0 2 1 0 1 0 7 6
10 24 18 12 49 23 19 42 11 15 :
CMHTEST pulmonary
CAPTION 'CMHTEST example 2',\
     !t('Data from Landis, Heyman & Koch (1978) Deaths from leukemia (LD)',\
        'observed at Atomic Bomb Casualty Commission'); STYLE=meta,plain
FACTOR  [NVALUES=60; LEVELS=5; LABELS=!t('0-9','10-19','20-34','35-49',\
        '50+')] Ages; !(12(1...5))
FACTOR  [NVALUES=60; LEVELS=2; LABELS=!t('LD','NLD')] Status; !((6(1,2))5)
FACTOR  [NVALUES=60; LEVELS=6; LABELS=!t('Not in city','0-9','10-49','50-99',\
        '100-199','200+')] Dose; !((1...6)10)
VARIATE [NVALUES=60] Deaths ; !(0,7,3,1,4,11,5015,10752,2989,694,418,387,5,4,\
        6,1,3,6,5973,11811,2620,771,792,820,2,8,3,1,3,9,5669,10828,2798,797,\
        596,624,3,19,4,2,1,10,6158,12645,3566,972,694,608,3,7,3,2,2,6,3695,\
        9053,2415,655,393,289)
CMHTEST [CLASS=Dose,Status,Ages] Deaths

Updated on March 8, 2019

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