Recent work has shown that there may be disadvantages in the use of the chi-square-like goodness-of-fit tests for the logistic regression model proposed by Hosmer and Lemeshow that use fixed groups of the estimated probabilities. A particular concern with these grouping strategies based on estimated probabilities, fitted values, is that groups may contain subjects with widely different values of the covariates. It is possible to demonstrate situations where one set of fixed groups shows the model fits while the test rejects fit using a different set of fixed groups. We compare the performance by simulation of these tests to tests based on smoothed residuals proposed by le Cessie and Van Houwelingen and Royston, a score test for an extended logistic regression model proposed by Stukel, the Pearson chi-square and the unweighted residual sum-of-squares. These simulations demonstrate that all but one of Royston's tests have the correct size. An examination of the performance of the tests when the correct model has a quadratic term but a model containing only the linear term has been fit shows that the Pearson chi-square, the unweighted sum-of-squares, the Hosmer-Lemeshow decile of risk, the smoothed residual sum-of-squares and Stukel's score test, have power exceeding 50 per cent to detect moderate departures from linearity when the sample size is 100 and have power over 90 per cent for these same alternatives for samples of size 500. All tests had no power when the correct model had an interaction between a dichotomous and continuous covariate but only the continuous covariate model was fit. Power to detect an incorrectly specified link was poor for samples of size 100. For samples of size 500 Stukel's score test had the best power but it only exceeded 50 per cent to detect an asymmetric link function. The power of the unweighted sum-of-squares test to detect an incorrectly specified link function was slightly less than Stukel's score test. We illustrate the tests within the context of a model for factors associated with low birth weight.

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Background: Physical activity (PA) is inversely associated with obesity but the effect has been difficult to quantify using questionnaires. In particular, the shape of the association has not yet been well described. Pedometers provide an opportunity to better characterize the association. Methods: Residents of households over the age of 25 years in randomly selected census districts in Tasmania were eligible to participate in the AusDiab cross-sectional survey conducted in 1999-2000. 1848 completed the AusDiab survey and 1126 of these (609 women and 517 men) wore a pedometer for 2-weekdays. Questionnaire data on recent PA, TV time and other factors were obtained. The outcomes were waist circumference (in cm) and body mass index (BMI) (kg/m 2 ). Results: Increasing daily steps were associated with a decline in the obesity measures. The logarithmic nature of the associations was indicated by a sharper decline for those with lower daily steps. For example, an additional 2000 steps for those taking only 2000 steps per day was associated with a reduction of 2.8 (95% confidence interval (CI): 2.1,4.4) cm in waist circumference among men (for women; 2.2 (95% CI: 0.6, 3.9 cm)) with a baseline of only 2000, steps compared to a 0.7 (95% CI 0.3, 1.1) cm reduction (for women; 0.6 (95% CI: 0.2, 1.0)) for those already walking 10 000 steps daily. In the multivariable analysis, clearer associations were detected for PA and these obesity measures using daily step number rather than PA time by questionnaire. Interpretation: Pedometer measures of activity indicate that the inverse association between recent PA and obesity is logarithmic in form with the greatest impact for a given arithmetic step number increase seen at lower levels of baseline activity. The findings from this study need to be examined in prospective settings.

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