EP 521, Spring 2007, Vol II, Part 6 Copyright © 2006 Trustees of the University of Pennsylvania 1 §8 Dose response and trends – an example to bridge contingency tables, ordinary linear regression, and logistic We have looked at measures of association with exposures (or treatments) that are unordered categories. These tests assume no special form in the relationship between exposure and outcome. They are “omnibus” tests for independence. Recall the eight Bradford-Hill (1965) criteria for evidence of causation (as contrasted with only association). (1) strength of association, (2) consistency of association in different populations, (3) specificity (cause leads to single effect), (4) temporality, (5) plausibility, (6) coherence (causal interpretation is consistent with biology of disease), (7) reversibility, and ... (8) Biological Gradient (or Dose Response) (See Rothman, pp. 24-7) EP 521 Spring 2006, Vol II, Part 6 2 Consider the 2 x k table, with each column having a proportion = ip (outcome probability or risk). We have already looked at 2 by 2 tables where012:Hpp=. Now consider situation where the alternative hypothesis is an ordering of the outcome variable produced by the k categories (of exposure). 1:...AkHpp≠≠. For ordered exposures there are more powerful tests of association available. By powerful, we mean that for a given sample size, one can demonstrate an effect with a narrower confidence interval (smaller p-value), or stated alternatively, it is easier to demonstrate an association if one truly exists. Also, if a trend is present, then it is more plausible that the association is consistent with causation (recall the Bradford Hill criteria). We will look at ways to demonstrate trend or dose response using: tables, graphs, and regression models.EP 521, Spring 2007, Vol II, Part 6 Copyright © 2006 Trustees of the University of Pennsylvania 3 Here is a table of the typical layout: Exposure Disease 0 1 2 3 4 5 Yes No The exposure scores: 0,...,5 are ordered but arbitrary. The cells (for count data) have frequencies We are interested in trends in proportions across the columns. 05...pp<<. EP 521 Spring 2006, Vol II, Part 6 4 Distinguish pairwise, “simultaneous”, and trend tests (1) Pairwise comparisons (adjacent categories, for example: 2 vs 1, 3 vs 2, 4 vs 3; or each category vs baseline: 1 vs 0, 2 vs 0, 3 vs 0, ... . Simple tests and estimates (OR, RR, IRR). These are all df=1 tests. If preplanned, these are appropriate, although must account for multiple tests (comparisons). But they do not explain entire picture of data. (2) Simultaneous comparisons (global, omnibus) Test whether there is an overall association between multiple category exposure and outcome. df=5 for the pictured example of 2 x 6 table. Always a good first step. Possible to have: larger pairwise p-values and small simultaneous p-value, pairwise p-values smaller than simultaneous p-value, no significant pairwise test but significant overall test (3) “Trend” is a special simultaneous test for increasing or decreasing relationship between risk and exposureEP 521, Spring 2007, Vol II, Part 6 Copyright © 2006 Trustees of the University of Pennsylvania 5 §8.1 Tables and graphs for trends Count data (risks) – Dose Response study Example: Oxford Childhood Cancer Study1Oxford Childhood Cancer Study (1953). Assn of childhood cancer and number of x-rays to mother. Case-control Number of prenatal maternal Xrays Totals 0 1 2 3 4 5 Control| 7673 | 239 | 154 | 65 | 28 | 29 | 8188 % | 51.14 | 45.44 | 43.63 | 40.37 | 32.18 | 30.85 | Case | 7332 | 287 | 199 | 96 | 59 | 65 | 8038 % | 48.86 | 54.56 | 56.37 | 59.63 | 67.82 | 69.15 | Col Tot 15005 526 353 161 87 94 16226 One can see from the table proportions (%) that as the x-ray dose increases, the fraction of cases increases steadily from 49% to 69%. One can plot the proportion of subjects who are cases against dose to see the relationship visually (a picture is worth a thousand words, and at least 10 to 20 “statistics”). 1Bithell JF, Steward MA. Prenatal irradiation and childhood malignancy: a review of British data from the Oxford Study. Br J Cancer. 1975;31:271-87. EP 521 Spring 2006, Vol II, Part 6 6 DOSE RESPONSE PLOT Cases as Prop. of total Prenatal Maternal x-rays The dose response slopes upwards and looks “linear” (or straight). We can test for non-linearity (later). There is no substitute for LOOKING AT THE DATA before one applies statistical methods. But we have more automated tools for testing trend. pNumber x-rays0 5.41EP 521, Spring 2007, Vol II, Part 6 Copyright © 2006 Trustees of the University of Pennsylvania 7 §8.2 Tests for trend (Armitage 1955) We have looked mostly at the 2χ statistic for which the degrees of freedom are (r-1)*(c-1). This statistic tests for overall association between the rows and the columns and assumes no ordering of either the rows or the columns. But in the dose response or trend case, with a binary outcome (case or control, disease or no disease), we have an ordered margin (either the row or the column, depending on how the data are set up). For dose response we need to look at a different chisq statistics for 2 by 6 table, a 2χ statistic with df=1. They have df=1 because they are like testing the slope of a regression line (we shall do this more explicitly later). EP 521 Spring 2006, Vol II, Part 6 8 Here are the data as entered into Stata: Case Expo Count 0 0 7673 0 1 239 0 2 154 0 3 65 0 4 28 0 5 29 1 0 7332 1 1 287 1 2 199 1 3 96 1 4 59 1 5 65 . tabodds y x [fw=count], graph -------------------------------------------------------------------------- x | cases controls odds [95% Conf. Interval] ------------+------------------------------------------------------------- 0 | 7332 7673 0.95556 0.92546 0.98664 1 | 287 239 1.20084 1.01145 1.42568 2 | 199 154 1.29221 1.04707 1.59473 3 |