Midterm Exam ANSWERS Categorical Data Analysis, CHL5407H

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1 Midterm Exam ANSWERS Categorical Data Analysis, CHL5407H 1. Data from a survey of women s attitudes towards mammography are provided in Table 1. Women were classified by their experience with mammography and by their belief in the value of mammography. A woman s belief in the value of mammography was measured by her answer to the question How likely is it that a mammogram could find a new case of cancer?. In their analyses of these data investigators obtained a p-value of less than using a Pearson chi-square test. (a) State the null and alternative hypotheses being evaluated using the Pearson chi-square test. The null hypothesis is that there is no association in the population between a woman s experience with mammography and her belief in the value of mammography. The alternative hypothesis is that there is an association in the population between a woman s experience with mammography and her belief in the value of mammography. (b) Is the statistical inference constructed using a Pearson chi-square test likely to be valid? Provide a brief explanation for your conclusions. Yes, this statistical inference is likely to be valid since the expected cell counts are all 5 or larger. Expected cell counts are provided in Appendix I. These were obtained by multiplying row and column totals and then dividing by 431, which is the number of study subjects. It is sufficient to show that the expected cell count for row 2, column 1 given by 94 37/431 = 8.1 > 5. All other expected cell counts must be larger since the second row and first column have the fewest marginal number of observations. (c) Suggest an alternative method which could be used to analyze these data. Give one advantage and one disadvantage of this procedure relative to the Pearson chi-square test. One alternative would be an exact test such as the extension of Fisher s exact test provided by SAS. Statistical inferences constructed using the Pearson chi-square test are approximate, their validity dependent on large sample theory and the central limit theorem. An advantage of an exact test is that its validity does not depend upon having large expected cell counts. A disadvantage of an exact test is that such procedures can be overly conservative. (Note that neither concern is particularly relevant for analyses of the data in Table 1 because (i) the expected cell counts are sufficiently large to support 1

2 use of the Pearson chi-square test and (ii) the observed association is highly statistically significant (p<0.001) using either an exact test or the Pearson chi-square test.) Logistic regression may also be used to analyse the data provided in Table 1. For example, one could model a women s experience with mammography as the study outcome grouping together women who had a mammography in the last year with women who also had a mammography but did so one or more years ago so that the subjects are divided into two groups those who ever had a mammogram and those who never had a mammogram. Note that the study outcome has been dichotomized which leads to some loss of information which is a disadvantage of the procedure. The study predictor measures a woman s belief in mammography. This variable could be modelled as either a quantitative covariate or using two dummy variables. In either case an advantage as compared to the Pearson chi-square test is that logistic regression provides estimates of the strength of the association between a women s belief in mammography and her experience with mammography while the former only provides an hypothesis test. (Extensions of logistic regression capable of modelling multinomial or ordinal outcomes are available. I will be mentioning these procedures later in the course.) (d) Do the data provided in Table 1 support investigators assertion that a woman s experience with mammography is associated with her belief in the value of mammography. (Note that I am asking you to summarize the information provided in Table 1. You do not need to construct a hypothesis test or a confidence interval). Yes, these data do offer very strong support for the investigators assertion that a woman s experience with mammography is associated with her belief in the value of mammography. This evidence may be seen by calculating the column percentages as are provided in Appendix I. For example, none of the women who believe mammograms are not likely to identify a breast cancer had a mammography within the past year. However 11% of women who believe mammograms are somewhat likely to identify a breast cancer had a mammography within the past year while 31% of women who believe mammograms are very likely to identify breast cancer had a mammogram within the past year. Thus women who have a stronger belief in the value of mammography were increasingly likely to have had a mammography in the past year. Results similarly consistent with the investigators assertion are seen using either the proportion of women who had a mammogram more than one year ago or women who never had a mammogram. 2. SAS code and output are provided in the Appendix for analyses of data from the low birth weight study. The variable LOW=1 for women who 2

3 had a low weight infant and LOW=0 for women who had a normal weight infant. The variable PTL measures the number of previous premature labors experienced by each study subject. (a) Estimate the probability that a mother who had 3 previous premature labors would have a normal weight infant using the results provided in the Appendix from fitting a logistic regression model. The probability that a mother who had 3 previous premature labors would have a normal weight infant is given by exp[ ( )] exp[ ( )] =0.19 using results from fitting a logistic regression model in Appendix II. Remember that the model is specified in terms of the probability of having a low weight infant. Therefore the estimated probability of having a normal weight infant is equal to one minus the estimated probability of having a low weight infant. (b) Do these data support the model assumption that there is a linear relationship between mother s history of premature labor and the logit of the risk of having a low weight infant? No, these data do not offer much support for the assumption that there is a linear relationship between mother s history of premature labor and the logit of the risk of having a low weight infant since the observed risk of having a low weight infant is neither strictly increasing nor strictly decreasing as a function of mother s history of premature labor. As is shown in Appendix II 26% (41/159) of mothers with no previous history of a premature labor had a low weight infant. This observed risk increases to 67% (16/24) for mother s who had one previous premature labor after which the observed risk of having a low weight baby fell to 40% (2/5) and 0% (0/1) for mothers who had, respectively, either two or three previous premature labors. (c) What can we conclude from these analyses about the association between a mother s history of premature labor and her risk of having a low weight infant? The estimated odds ratio for having a low weight infant is 2.2 with a 95% confidence interval of (1.2,4.2). Thus the logistic regression model suggests that a mother s risk of having a low weight infant increases with the number of premature labors she has had and that this association is statistically significant (p=0.0115). These conclusions depend, however, on the assumption that there is a linear relationship between mother s history of premature labor and the logit of the risk of having a low weight infant. As discussed above this assumption has little support from the data. On the other hand there are only six women with a history of two or three premature labors. Thus presumably the observed statistically significant association is likely 3

4 primarily a consequence of the sharp increase in the risk of having a low weight baby comparing women with one premature labor to women with zero premature labors. Finally, any conclusions we reach about the association between a mother s history of premature labor and her risk of having a low weight infant must be seen as preliminary since we have made no attempt to account for confounding variables or sources of effect modification. For example, no distinction is made between the number of premature labors and the number of times a mother has given birth. Thus mother s giving birth for the first time are treated as identical to mother s who have given birth many times but have never had a premature labor. 4

5 Appendix I data x1; input mammog_exper $ how_likely $ n; cards; never not 32 never somewhat 73 never very 129 plus1 not 5 plus1 somewhat 20 plus1 very 69 within1 not 0 within1 somewhat 12 within1 very 91 ; proc freq;tables mammog_exper*how_likely / chisq expected nopercent norow; weight n; mammog_exper Frequency how_likely Expected Col Pct not somewhat very Total never plus within Total Statistics for Table of how_likely by mammog_exper Statistic DF Value Prob Chi-Square <

6 Appendix II data x1;infile "c:\nklar\chl5407h\data\bwt\bwt.dat"; input id low age lwt race smoke ptl ht ui ftv bwt; proc freq;table low*ptl; proc logistic descending;model low=ptl; TABLE OF LOW BY PTL LOW PTL Frequency (Col Pct) Total (74.21) (33.33) (60.00) (100.00) (25.79) (66.67) (40.00) (0.00) Total Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio Score Wald Analysis of Maximum Likelihood Estimates Standard Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 ptl Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits ptl