> chisq.test(bigdat$marital[bigdat$awwave == 1], bigdat$studygrp[bigdat$awwave == 1])$expected
                                  bigdat$studygrp[bigdat$awwave == 1]
bigdat$marital[bigdat$awwave == 1]  1 VUBook 2 CommercialBook 3 NoBookGiven
                           Single  48.467005        51.593909     53.939086
                           Married 10.385787        11.055838     11.558376
                           Other    3.147208         3.350254      3.502538
Warning message:
In chisq.test(bigdat$marital[bigdat$awwave == 1], bigdat$studygrp[bigdat$awwave ==  :
  Chi-squared approximation may be incorrect

03/02/09

02/27/09

Descriptive Statistics by Study group (experimental condition)

+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|                           |N  |1 VUBook           |2 CommercialBook   |3 NoBookGiven      |  Test                       |
|                           |   |(N=62)             |(N=66)             |(N=70)             |Statistic                    |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|income : 1 Less than $8,000|120|           29% (13)|           29% (10)|           15% ( 6)|Chi-square=7.2 d.f.=2 P=0.027|
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    2 $8,000 - $12,000     |   |           22% (10)|           15% ( 5)|           20% ( 8)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    3 $12,001 - $16,000    |   |           11% ( 5)|           18% ( 6)|            2% ( 1)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    4 $16,001 - $21,000    |   |            4% ( 2)|           15% ( 5)|           12% ( 5)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    5 $21,001 - $26,000    |   |           13% ( 6)|            6% ( 2)|           12% ( 5)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    6 $26,001 - $30,000    |   |            4% ( 2)|            3% ( 1)|            5% ( 2)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    7 $30,001 - $40,000    |   |            4% ( 2)|            9% ( 3)|           10% ( 4)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    8 $40,001 - $50,000    |   |            7% ( 3)|            0% ( 0)|            7% ( 3)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|    over $50,000           |   |            4% ( 2)|            6% ( 2)|           17% ( 7)|                             |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+
|as.numeric(income)         |120|            1/2/5  |            1/3/4  |            2/5/7  |   F=3.6 d.f.=2,117 P=0.031  |
+---------------------------+---+-------------------+-------------------+-------------------+-----------------------------+

02/26/09

02/25/09

ANOVA (Row Mean Scores) Statistic

The ANOVA statistic can be used only when the column variable Y lies on an ordinal (or interval) scale so that the mean score of Y is meaningful. For the ANOVA statistic, the mean score is computed for each row of the table, and the alternative hypothesis is that, for at least one stratum, the mean scores of the R rows are unequal. In other words, the statistic is sensitive to location differences among the R distributions of Y.

The matrix of column scores C_h has dimension 1 ×C, the column scores are determined by the SCORES= option.

The matrix of row scores R_h has dimension (R-1) ×R and is created internally by PROC FREQ as

    R_h = [ I_{R-1} , -J_{R-1} ] 

where I_{R-1} is an identity matrix of rank R-1, and J_{R-1} is an (R-1) ×1 vector of ones. This matrix has the effect of forming R-1 independent contrasts of the R mean scores.

When there is only one stratum, this CMH statistic is essentially an analysis of variance (ANOVA) statistic in the sense that it is a function of the variance ratio F statistic that would be obtained from a one-way ANOVA on the dependent variable Y. If nonparametric scores are specified in this case, then the ANOVA statistic is a Kruskal-Wallis test.

If there is more than one stratum, then this CMH statistic corresponds to a stratum-adjusted ANOVA or Kruskal-Wallis test. In the special case where there is one subject per row and one subject per column in the contingency table of each stratum, this CMH statistic is identical to Friedman's chi-square. See Example 2.8 for an illustration.