Stat 642, Lecture notes for 04/12/05 96


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1 Stat 642, Lecture notes for 04/12/05 96 HosmerLemeshow Statistic The HosmerLemeshow Statistic is another measure of lack of fit. Hosmer and Lemeshow recommend partitioning the observations into 10 equal sized groups according to their predicted probabilities. Then where G 2 HL = 10 j=1 O j E j 2 E j 1 E j /n j χ2 8 n j = Number of observations in the j th group O j = i E j = i y ij = Observed number of cases in the j th group ˆp ij = Expected number of cases in the j th group Example: Using the THS data described earlier 4/7/2005, I added the lackfit option to the model statement in PROC LOGISTIC as follows: model cens28dy = trt age sbpcat aist hctbase / lackfit; The output corresponding to the HosmerLemeshow statistic is: Partition for the Hosmer and Lemeshow Test CENS28DY = 0 CENS28DY = 1 Group Total Observed Expected Observed Expected Hosmer and Lemeshow GoodnessofFit Test ChiSquare DF Pr > ChiSq
2 Stat 642, Lecture notes for 04/12/05 97 This output shows the observed and expected numbers of cases and controls within each group, and the final test statistic. The chisquare statistic suggests that there may be lack of fit, so I refit the model including a quadratic term for age: model cens28dy = trt age age sbpcat aist hctbase / lackfit; Partition for the Hosmer and Lemeshow Test CENS28DY = 0 CENS28DY = 1 Group Total Observed Expected Observed Expected Hosmer and Lemeshow GoodnessofFit Test ChiSquare DF Pr > ChiSq This model shows no evidence of lack of fit based on the HL statistic, so apparently any lack of fit has been corrected. Ideally, incorrect model specification such as nonlinearity in the predictors or missing predictors should be detectable by this statistic. However, depending on the model, G 2 LH may not be particularly sensitive to departures from the true model. The HL test does not provide any information regarding the nature of the lack of fit observed. Other procedures may be more useful. Other goodnessoffit procedures Tsiatis A note on the goodnessoffit test for the istic regression model, Biometrika, 67: , 1980 proposes a test for goodnessoffit obtained by grouping observations into k groups according to covariate values. The goodnessoffit test uses the score test of H 0 that the fitted model is correct, versus the model which adds to the fitted model a categorical variable corresponding to the groupings. For example, for the model above, one could group observations by age categories possibly equally spaced, or of equal size  the groups are somewhat arbitrary, and perform the score test for the addition of these groupings to the model similar to what one would do in a forward selection procedure.
3 Stat 642, Lecture notes for 04/12/05 98 Lin, Wei and Ying Model checking techniques based on cumulative residuals, Biometrics, 58:112, 2002 propose assessing goodnessoffit using moving sums of residuals of the form: W x = i Ix b < X ij < x + by i Ey i where I is one if the argument is true, and zero otherwise, x is an arbitrary value of covariate j and X ij is the value of covariate j for subject i. W x is simply the total observed  expected for all observations with the value covariate j within b units of x. By plotting W x against x for values of x over the range of the covariate, patterns may be apparent which suggest lack of fit. LWY propose a formal test using simulation to assess whether the observed patterns are likely by chance alone. The score test of Tsiatis may also be used to provide a formal test. Polychotomous Responses note that in the interest of time, I skipped over some of the material in this section during class Now we consider the case in which we have more than two levels of outcome variable. Examples Type of Disease Site of Onset e.g. tumor Disease Severity ordered Cause of Death Suppose that we have k + 1 response categories, 0, 1,..., k. Under a multinomial model there are k + 1 event probabilities which sum to one. Hence there are k independent its to model. We consider three approaches to this modeling, all of which produce different results. Loglinear model Continuation Ratios Cumulative Logit Loglinear Model Poisson Sampling model Suppose we have a covariate with levels 0, 1, 2,.... We may construct the following table:
4 Stat 642, Lecture notes for 04/12/05 99 Response k Covariate Values For the i, j cell, we have the linear model λ ij = α i + β j + γ ij where we impose the constraint that γ 0j = γ i0 = 0 i, j. We could, for example, choose a baseline category for the response, say i = 0. Then λij λ 0j = pij p 0j = α i α 0 + γ ij Then γ ij is the odds ratio for response level i versus level 0 and exposure level j versus level 0. This may be called the baseline it model. Similarly, if the responses are naturally ordered, then it may be reasonable to consider adjacent categories: λi+1,j λ i,j = pi+1,j p i,j = α i+1 α i + γ i+1,j γ i,j = α i + γ i,j Then γij is the odds ratio for response level i + 1 versus level i and exposure level j versus level 0. More generally, we might have In which case pij p 0j λ ij = x T j β i We may compute the probabilities themselves via = x T j β i β 0 = x T j β i. p ij = e x jβ i 1 + l 1 ex jβ l For example, if k = 2, we have: p 0j = e x jβ1 + e x jβ2, p 1j = e x jβ e x jβ 1 + e x jβ 2 p 2j = e x jβ e x jβ 1 + e x jβ 2 Hypothetical example: Suppose we consider death from Heart Disease or Cancer. There is no particular order to these outcomes, so this kind of model may make some sense. In the accompanying
5 Stat 642, Lecture notes for 04/12/ plot, the probabilities of death from heart disease and cancer change as a function of age and this model allows them to change quite independently. Hypothetical Example Death from Heart Disease vs. Cancer Heart Disease Cancer Survival or Other Probability Age Note that if we consider a plot of its for each of heart disease and cancer versus survival we have straight lines: Hypothetical Example  it scale Heart Disease Cancer p i /p 0 Age
6 Stat 642, Lecture notes for 04/12/ Continuation Ratios Take k = 2 for a moment and consider two parallel schemes for constructing the 2 independent its: In General: Scheme A p0j p 1j + p 2j p1j p 2j pij l>i p lj Scheme B p2j p 0j + p 1j p1j p 0j pij l<i p lj If the categories are 0, 1, 2,..., k, then under Scheme A, we first consider the lowest category i = 0 versus all remaining categories. For each category i, i < k, we consider category i versus all higher categories. Under Scheme B, we consider for each category i, i < 0, we consider category i versus all lower categories. For example, if the categories are Alive, Heart Disease Death and Cancer Death, we might consider its for Alive versus Dead and among the deaths Heart Disease Death versus Cancer Death. If we set the i th it equal to x T j β i, we can express the multinomial probabilities of being in a particular category as follows: take k = 3, Scheme A. First we have that next, Similarly, Finally, p 0j = Pr{y j = 0} = ext β e xt β 0 p 1j = Pr{y j = 1} = Pr{y j = 1 y j 1} Pr{y j 1} = e xt β e xt β e xt β 0. p 2j = Pr{y j = 2} = Pr{y j = 2 y j 2} Pr{y j 2 y j 1} Pr{y j 1} = e xt β e xt β e xt β e xt β 0 p 3j = Pr{y j = 3} = Pr{y j = 3 y j 2} Pr{y j 2 y j 1} Pr{y j 1} = e xt β e xt β e xt β 0
7 Stat 642, Lecture notes for 04/12/ In all cases, the p ij are products of conditional probabilities each of which depends one only on one of the β i. Since the likelihood is composed of products of the above p ij, it will also factor into a product of conditional likelihoods, each of which can be independently maximized. Hence, this model can be fit as a series of binary istic regression models. Each binary model is fit by restricting the data set to only those observations with responses in categories i through k under Scheme A or 0 through i under Scheme B and considering response level i versus the remaining categories. Continuation Ratio models give different fitted values than linear models and Scheme A gives different fitted values than Scheme B. Cumulative Logits. Cumulative Logit models are typically used when we have ordinal responses. That is, we may consider the outcomes as a series of progressively worse or more extreme outcomes. In this case the relevant questions usually involve probabilities of subjects progressing from one category to the next, or probabilities of a subject being above of a given category versus at or lower than a given category. Example: Disease Severity Percent of Subjects None Mild Moderate Severe  + Exposure In the above plot, the size of the none category decreases slightly with exposure. This suggests that exposure increases the risk of disease without regard for severity. On the other hand, the mild category also decreases with exposure, yet this fact by itself is not very meaningful since we don t know whether or not subjects who otherwise would have been in the mild category were moved to the none category or the moderate or worse categories. The conclusion might be quite different depending on which of these happened. More meaningful is the observation that
8 Stat 642, Lecture notes for 04/12/ the combined none and mild categories shrunk, suggesting that the rate of moderate or worse disease increased. This leads us to the cumulative it model: Pr{yj i} = α i + x T j Pr{y j > i} β i. If β i = β i, this is the Proportional Odds model. We also require that α 0 α 1... although this will hold automatically in a proportional odds model. eα i+x T j β i We have Pr{y j i} =. 1 + e α i+x T j β i Note that if β i β i for some i i, then α i + β i x = α i + β i x for some x and the lines will cross. If α i + x T j β i > α i + x T j β i for some i < i then Pr{y j i} > Pr{y j i } which is nonsense. Hence if a cumulative it model is NOT a proportional odds model, there may be particular values of x for which the predicted values are inconsistent. Under the proportional odds model, the shapes of all curves are the same, they are shifted progressively to the left for increasing i. The horizontal distance between adjacent curves is α i α i 1 /β. Pr{y i} i = 0 i = 1 i = 2 Pr{y = 3 x} Pr{y = 2 x} Pr{y = 1 x} Pr{y = 0 x} Odds Ratios: Let the odds of {y j i} be ψ ij = Pr{y j i} Pr{y j > i} = eα i+x T j β x so and ψ ij ψ i j ψ ij ψ ij = e α i α i is independent of j proportional odds = e x j x j T β is independent of i With more than two levels of the outcome variable, SAS fits a proportional odds model and uses the score test for to test for proportionality.
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