CDA Chapter 3 part II

Transcription

1 CDA Chapter 3 part II

2 Two-way tables with ordered classfications Let u 1 u 2... u I denote scores for the row variable X, and let ν 1 ν 2... ν J denote column Y scores. Consider the hypothesis H 0 : X and Y are independent VS H 1 : Y is a linear function of X Mantel-Haenszel (MH) statistic M 2 = (n 1)r 2 where r is the Pearson correlation coefficient between X and Y based on the above scores. For large sample, M 2 is approximately chi-squared with df = 1.

3 Example: Is happiness associated with political ideology? With scores (1,2,3) for each variable, the correlation is r = The linear trend test statistic M 2 = (321 1)(0.135) 2 = 5.85 This shows strong evidence of association (P = 0.016)

4 Sensitivity to choice of scores Cochran (1954) noted that... If the set of scores is poor, in that it badly distorts a numerical scale that really does underlie the ordered classification, the test will not sensitive For most data sets, different choices of monotone scores give similar results Scores that are linear transforms of each other, such as (1,2,3,4) and (0,2,4,6), have the same absolute correlation and hence the same M 2. Results may depend on the scores, when data are highly unbalanced, with some categories having many more observations than others. It is usually better to select scores that reflect perceived distances between categories.

5 Sensitivity to choice of scores Equally spaced scores often provide a reasonable compromise when categorical labels do not suggest obvious choices When unsure, do sensitivity analysis and check whether results are similar If you choose a set of scores, and get a significant result, then this suggests that row and column are not independent. However, you shouldn t keep choosing a lot of different scores until you get a significant result.

6 Example: Infant birth defects by maternal alcohol consumption With Y score {1, 2, 3, 4, 5}, M 2 = 1.83 and P = 0.18 With Y score as midranks {(1 + 17, 114)/2, 24, 365.5, 32, 013, 32, 473, 32, 555.5}, M 2 = 0.35 and P = 0.55 With Y score {0, 0.5, 1.5, 4.0, 7.0}, M 2 = 6.57 and P = 0.01

7 Monotone trend alternatives to independence Consider the hypothesis: H 0 : X and Y are independent VS H 1 : Y is a monotone function of X Gamma test statistic: z = ˆγ/SE where SE is the standard error of ˆγ drived based on the delta method. Under H 0, z follows a standard normal distribution. For the example on happiness and political ideology, ˆγ = 0.185, z = 0.185/0.078 = 2.37 and the two sided P = An approximate 95% CI for γ is ± 1.96(0.078) = (0.032, 0.338) The true association seems to be relatively weak.

8 Extra power with ordinal tests Consider the same example of happiness and political ideology, Ignoring the ordering of the categories, Pearson chi-squared statistics for testing independence is MH test is X 2 = 7.07 with df = 4, P = 0.13 M 2 = 5.85 with df = 1, P = Gamma test z = 2.37, P = The latter two are ordinal tests which have more power in this example because they are designed to detect linear or monotone patterns, whereas the X 2 and G 2 refer to the most general alternative, whereby cell probabilities exhibit any type of statistical dependence.

9 Trend tests for 2XJ tables Using scores for the Y variable, contruct the MH test, which detect differences between the two row means of the scores on Y With midrank scores for Y, the MH test is also called Wilcoxon or Mann-Whitney test. It is two-sample t-test with ranks. The MH test is also equivalent to the test based on z = C D SE 0 where C and D are numbers of concordant and discordant pairs respectively. Find score CI for the measure = P(Y 1 > Y 2 ) P(Y 2 > Y 1 )

10 Nominal X ordinal tables and IX2 tables Extension to a nominal row variable with more than two categories, the Mann-Whitney test extends to Kruskal Wallis test which is ANOVA test on ranks of the Y values. In IX 2 tables, Y is binary. The linear trend statistic then refers to a linear trend in the probability of either response category, such as the probability of malformation as a function of alcoho consumption. The test in this case, often called Cochran-Armitage trend test, which is also related to logistic regression.

11 Small-sample inference for contingency tables

12 Small-sample inference for contingency tables Conditional on the row and column margins, the probability of all possible tables. Let θ be the odds ratio. Consider the one sided hypothesis H 0 : θ = 1 vs H 1 : θ > 1 The P-value equals the sum of the probabilities of the tables that have large n 11, therefore. P value =

13 Fisher s exact test for 2x2 tables Conditioning on both sets of marginal totals, Fisher s exact test statistic is the table probability

14 Fisher s exact test for 2x2 tables For the one-sided alternative, the same P-value results using tables ordered according to larger n 11, larger odds ratio, or larger difference of proportions. For a two-sided alternative, the most common approach sums probabilities of equally or less likely tables, that is, P value = P(p(n 11 ) p(t o )) for the observed value t o In the previous example, the two sided P-value is =