I4C Pooled Dataset: Problems and Solutions Stan Lemeshow & Gary Phillips The Ohio State University

1. I4C Pooled Dataset:Problems and SolutionsStan Lemeshow & Gary PhillipsThe Ohio State University • We joined the project at the beginning of 2013 • We noted interesting challenges: • 6 non-homogeneous cohorts • How best to identify suitable control group (10% sample or everyone)? • If 10% sample, should we use Prentice weights? • should we use stratified analyses or random effects models? • How to incorporate birth weight in models? • dichotomous, polychotomous, or continuous • How to deal with large amount of missing data? • statistical packages drop entire case when any of the variables are missing • How to deal with confounding and effect modification? • How to overcome, for statistical analyses, small numbers of cancers in some of the birth weight groups? • How to assess and incorporate appropriate scale of continuous covariates?

2. • 6 non-homogeneous cohorts a: based on a random, representative, 10% sample of the entire cohort b: based on neonates with data on gestational age ALSPAC: Avon Longitudinal Study of Parents and Children (UK) CPP: The Collaborative Perinatal Project (USA) DNBC: Danish National Birth Cohort (Denmark) JPS: Jerusalem Perinatal Study (Israel) MoBa: Norwegian Mother and Child Cohort Study (Norway) THIS: Tasmanian Infant Health Survey (Australia) Issue: Is it possible to create a set of weights to reflect some overarching population? • We believe that there really isn’t an overarching population so use of specially chosen statistical weights is unnecessary.

3. Oversampling Cox stratified regression using Prentice Weights using 10% of the controls Stratified Cox regr. -- Breslow method for ties No. of subjects = 1998 Number of obs = 1998 No. of failures = 376 Time at risk = 18458.09313 LR chi2(3) = 5.44 Log likelihood = -39727.245 Prob > chi2 = 0.1421  ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- birthwt_cat | 0 | 0.79 0.21 -0.89 0.375 0.46 1.34 2 | 1.22 0.16 1.47 0.142 0.94 1.59 | gestage | 0.95 0.02 -2.13 0.033 0.90 1.00 weight | 1.00 (offset) ------------------------------------------------------------------------------ Stratified by study Cox stratified regression without using Prentice Weights using 10% of the controls Stratified Cox regr. -- Breslow method for ties No. of subjects = 1998 Number of obs = 1998 No. of failures = 376 Time at risk = 18458.09313 LR chi2(3) = 3.79 Log likelihood = -2175.7188 Prob > chi2 = 0.2851  ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- birthwt_cat | 0 | 0.81 0.22 -0.79 0.429 0.47 1.37 2 | 1.17 0.16 1.16 0.244 0.90 1.53 | gestage | 0.96 0.02 -1.80 0.071 0.91 1.00 ------------------------------------------------------------------------------ Stratified by study Prentice weights are used to adjust the analysis when the cases are oversampled in the analysis. We found that if we used Prentice weights in the Cox regression the estimated hazard ratios were essentially the same as the hazard ratios when Prentice weights were not used. we decided not to use them. • Original plan was to take all cases and 10% of controls • Prentice weights have been proposed for handling the oversampling • If the plan is to use 10% of controls in all cohorts, should we use Prentice weights or not? No need to use Prentice weights

4. Cox stratified regression without using Prentice Weights using 10% of the controls Stratified Cox regr. -- Breslow method for ties No. of subjects = 1998 Number of obs = 1998 No. of failures = 376 Time at risk = 18458.09313 LR chi2(3) = 3.79 Log likelihood = -2175.7188 Prob > chi2 = 0.2851  ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- birthwt_cat | 0 | 0.81 0.22 -0.79 0.429 0.47 1.37 2 | 1.17 0.16 1.16 0.244 0.90 1.53 | gestage | 0.96 0.02 -1.80 0.071 0.91 1.00 ------------------------------------------------------------------------------ Stratified by study Cox stratified regression without using Prentice Weights and using all observations Stratified Cox regr. -- Breslow method for ties No. of subjects = 111430 Number of obs = 111430 No. of failures = 376 Time at risk = 1101297.732 LR chi2(3) = 4.41 Log likelihood = -3511.1489 Prob > chi2 = 0.2203 ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- birthwt_cat | 0 | 0.86 0.23 -0.58 0.565 0.50 1.46 2 | 1.20 0.16 1.37 0.170 0.92 1.57 | gestage | 0.95 0.03 -1.85 0.064 0.90 1.00 ------------------------------------------------------------------------------ Stratified by study Use all observations or only10% of Controls? Initial analyses used 10% of non-cases vs. all cases from each of the 6 cohorts. We thought it made more statistical sense use all of the observations if the goal of this study was to find the true relationship between birth weight and childhood cancer. Presumably this relationship would be independent of the overarching population. Since we are not trying to model a specific population using the 6 cohorts sampling weights would not be needed. Sampling weights are applied in analyses so that the results (prevalence of childhood cancer) mirror an overarching population the study sample does not reflect. No need to use only a 10% sample

5. Cox stratified regression without using Prentice Weights and using all observations Stratified Cox regr. -- Breslow method for ties No. of subjects = 111430 Number of obs = 111430 No. of failures = 376 Time at risk = 1101297.732 LR chi2(3) = 4.41 Log likelihood = -3511.1489 Prob > chi2 = 0.2203  ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- birthwt_cat | 0 | 0.86 0.23 -0.58 0.565 0.50 1.46 2 | 1.20 0.16 1.37 0.170 0.92 1.57 | gestage | 0.95 0.03 -1.85 0.064 0.90 1.00 ------------------------------------------------------------------------------ Stratified by study Random-effects (shared frailty) Cox regression without using Prentice Weights and using all observations Cox regression -- Breslow method for ties Number of obs = 111430 Gamma shared frailty Number of groups = 6 Group variable: study No. of subjects = 111430 Obs per group: min = 8571 No. of failures = 376 avg = 18571.67 Time at risk = 1101297.732 max = 49334 Wald chi2(3) = 4.59 Log likelihood = -4047.5665 Prob > chi2 = 0.2042 ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- birthwt_cat | 0 | 0.85 0.23 -0.61 0.545 0.50 1.45 2 | 1.21 0.16 1.42 0.156 0.93 1.58 | gestage | 0.95 0.03 -1.83 0.067 0.90 1.00 -------------+---------------------------------------------------------------- theta | 1.15993 .5899787 ------------------------------------------------------------------------------ Likelihood-ratio test of theta=0: chibar2(01) = 439.68 Prob>=chibar2 = 0.000 Note: standard errors of hazard ratios are conditional on theta. Stratified analyses vsrandom effects models Sensitivity analysis: • compare results of stratified Cox regression to a random-effects (shared frailty) Cox regression. • The hazard ratios very similar in both regressions. • The cohort identifier was the stratification variable and was also the random-effects term. use stratified analysis as it runs much faster on the computer compared to the random-effects analysis.

6. Switchingfrom a 3-level birth weight to a 2-level birth weight variable Childhood | Birth weight cancer | < 2500 2500 to < >= 4000 | Total -----------+---------------------------------+---------- 0. No | 8,509 92,397 11,030 | 111,936 1. Yes | 20 282 75 | 377 -----------+---------------------------------+---------- Total | 8,529 92,679 11,105 | 112,313 Childhood | Birth weight leukemia | < 2500 2500 to < >= 4000 | Total -----------+---------------------------------+---------- 0. No | 8,527 92,592 11,079 | 112,198 1. Yes | 2 87 26 | 115 -----------+---------------------------------+---------- Total | 8,529 92,679 11,105 | 112,313 Childhood | Birth weight ALL | < 2500 2500 to < >= 4000 | Total -----------+---------------------------------+---------- 0. No | 8,527 92,605 11,083 | 112,215 1. Yes | 2 74 22 | 98 -----------+---------------------------------+---------- Total | 8,529 92,679 11,105 | 112,313 • When we analyzed birth weight using a 3-level variable • <2,500 g • 2,500 to < 4,000 g • ≥ 4,000 g the regression was unstable since the Leukemia and ALL outcomes only had only 2 babies born weighing < 2,500 grams. • There were only 4 observations in the leukemia and ALL groups when we looked at the lowest 10 percentile of birth weight. • Thus we dichotomized birth weight at: • 4,000 g • 3,500 g • 3,000 g • top 10 percentile within each of the 6 cohorts • We also analyzed birth weight using a continuous variable

7. Dealing with a large amount of missing data • Statistical packages drop the entire case when any of that case’s variables are missing • Solution: Multiple imputation (MI) • We used chained MI to impute 20 complete datasets. • Note that the choice to use 20 imputed complete datasets was based on our collective experience running MI. • After generating the 20 datasets proportional hazard Cox regression was performed separately on each imputation m = 1, … , 20 and the results are pooled into a single multiple-imputation result. • The “chained” method fills in missing values in multiple variables iteratively by using chained equations, a sequence of univariate imputation methods with fully conditional specification (FCS) of prediction equations. • It accommodates arbitrary missing-value patterns. Specifically we used: • truncated linear regression for continuous variables (paternal age, maternal height, pregnancy weight change, and pre-pregnancy BMI) where the imputations are limited to lower and upper boundaries set at the minimum and maximum values of the non-missing observations of a particular continuous variable. • logistic regression for dichotomous variables (first born and maternal smoking). • predictor variables used to impute the missing data were maternal age, gestational age, birth weight, sex of child, and cohort.

8. Table of missing observations by cohort

9. Summary of continuous variables needed to set the boundaries on the truncated regression used in the MI variable | N mean sd min max -----------------+-------------------------------------------------- pat_age | 96678 29.8 6.6 14.0 71.9 mat_height | 99173 162.9 7.2 101.6 203.2 tot_pregwtchge | 95512 11.4 5.6 -76.0 68.0 mat_prepreg_BMI | 95562 22.9 4.1 11.0 61.0 -------------------------------------------------------------------- Multiple Imputation

10. Confounding and Effect Modification • We checked all variables in the dataset to determine if any of them confounded the relationship between cancer and dichotomous birth weight or continuous birth weight. • A variable is potentially a confounder when its addition to the model with only birth weight changed the birth weight coefficient by more than 15% in either direction. This has nothing to do with statistical significance. • Assessment of confounding was made prior to employing multiple imputation. • we used a set of observations that was common to both regressions (with and without the confounder). • This produced the following set of possible univariableconfounders: • gestational age • maternal age • paternal age • maternal height • pregnancy weight change • pre-pregnancy BMI • first born • any maternal smoking • This same set of confounders was produced for all 5 definitions of birth weight.

11. Effect Modification Table: Outcome is cancer where β1 is the coefficient for ≥ 4,000 and β2 is the coefficient for a 1 kg increase in continuous birth weight Note: numbers of observations in crude and adjusted models are identical • A covariate that had a statistically significant interaction (p≤ 0.05) with birth weight was considered to be an effect modifier. • When we checked none of the covariates were considered effect-modifiers (all p-values were quite large). • This was true for all 5 definitions of birth weight. • Note: Had a variable been an effect modifier, it could not also have been a confounder.

12. Stratified Cox regr. -- Breslow method for ties No. of subjects = 105494 Number of obs = 105494 No. of failures = 342 Time at risk = 1062773.443 LR chi2(3) = 1.80 Log likelihood = -3182.6531 Prob > chi2 = 0.6157 --------------------------------------------------------------------------------------------- _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] ----------------------------+---------------------------------------------------------------- 1.birthwt_cat4000 | 0.21 0.15 1.35 0.178 -0.09 0.51 1.mat_smk_any | 0.07 0.14 0.49 0.625 -0.20 0.34 | birthwt_cat4000#mat_smk_any | 1 1 | -0.23 0.36 -0.65 0.515 -0.93 0.47 --------------------------------------------------------------------------------------------- Example of checking for effect modification with categorical birth weight Example of checking for effect modification with continuous birth weight for a 1 kg increase Stratified Cox regr. -- Breslow method for ties No. of subjects = 105494 Number of obs = 105494 No. of failures = 342 Time at risk = 1062773.443 LR chi2(3) = 2.09 Log likelihood = -3182.5069 Prob > chi2 = 0.5541 ------------------------------------------------------------------------------------------ _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------------------+---------------------------------------------------------------- birthwt_kg | 0.16 0.12 1.36 0.175 -0.07 0.39 1.mat_smk_any | 0.28 0.72 0.39 0.696 -1.14 1.70 | mat_smk_any#c.birthwt_kg | 1 | -0.07 0.21 -0.34 0.736 -0.48 0.34 ------------------------------------------------------------------------------------------

13. Assessing the Scale of the Continuous Covariates From the multivariable fractional polynomials we can see that paternal age is not linear in the logit Deviance for model at the end of cycle 1=5344.807 , 74641 observations Variable Model (vs.) Deviance Dev diff. P Powers (vs.) ---------------------------------------------------------------------- mat_height lin. FP2 5344.807 0.703 0.872 1 3 3 Final 5344.807 1 tot_pregw... lin. FP2 5344.807 6.510 0.089 1 1 1 Final 5344.807 1 mat_prepr... lin. FP2 5344.807 4.215 0.239 1 -2 -2 Final 5344.807 1 mat_age lin. FP2 5344.807 0.864 0.834 1 0 0 Final 5344.807 1 pat_age lin. FP2 5357.821 13.014 0.005+ 1 3 3 FP1 5353.602 8.794 0.012+ 3 Final 5344.807 3 3 [birthwt_cat4000 included with 1 df in model] gestage lin. FP2 5344.807 0.191 0.979 1 -2 -2 Final 5344.807 1 Fractional polynomial fitting algorithm converged after 2 cycles. Note: This is an iterative process that, after 5 or 6 cycles, produces the final results on the next slide. • We checked the scale of the continuous covariates and birth weight to ensure that they were linear in the log-hazard • The method of fractional polynomials was used for this purpose. • All continuous variables were determined to be linear and we did not need to transform any of them.

14. Final multivariable fractional polynomial model for _t -------------------------------------------------------------------- Variable | -----Initial----- -----Final----- | df Select Alpha Status df Powers -------------+------------------------------------------------------ birthwt_c... | 1 1.0000 0.0500 in 1 1 mat_age | 4 1.0000 0.0500 in 1 1 pat_age | 4 1.0000 0.0500 in 4 3 3 mat_height | 4 1.0000 0.0500 in 1 1 tot_pregw... | 4 1.0000 0.0500 in 1 1 mat_prepr... | 4 1.0000 0.0500 in 1 1 gestage | 4 1.0000 0.0500 in 1 1 -------------------------------------------------------------------- Next we run fractional polynomials where we log all 44 transformations and notice that a quadratic transformation is very close to the complicated (3,3) Model # Deviance Power 1 Power 2 1 5359.042 -2.0 . 2 5359.114 -1.0 . : : : : 37 5345.886 2.0 0.5 38 5345.164 3.0 0.5 39 5346.321 1.0 1.0 40 5345.497 2.0 1.0 41 5344.866 3.0 1.0 42 5344.859 2.0 2.0 43 5344.404 3.0 2.0 44 5344.111 3.0 3.0

15. Stratified Cox regr. -- Breslow method for ties No. of subjects = 94661 Number of obs = 94661 No. of failures = 348 Time at risk = 975210.3499 LR chi2(3) = 6.00 Log likelihood = -3215.7098 Prob > chi2 = 0.1116 --------------------------------------------------------------------------------- _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] ----------------+---------------------------------------------------------------- birthwt_cat4000 | 1.14 0.16 0.91 0.362 0.86 1.50 gestage | 0.98 0.03 -0.86 0.391 0.93 1.03 pat_age | 0.98 0.01 -2.18 0.029 0.96 1.00 --------------------------------------------------------------------------------- Stratified by study Stratified Cox regr. -- Breslow method for ties No. of subjects = 94661 Number of obs = 94661 No. of failures = 348 Time at risk = 975210.3499 LR chi2(4) = 7.32 Log likelihood = -3215.0512 Prob > chi2 = 0.1201 --------------------------------------------------------------------------------- _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] ----------------+---------------------------------------------------------------- birthwt_cat4000 | 1.13 0.16 0.89 0.375 0.86 1.49 gestage | 0.98 0.03 -0.86 0.392 0.93 1.03 pat_age | 1.07 0.08 0.83 0.409 0.91 1.25 pat_age2 | 1.00 0.00 -1.11 0.269 1.00 1.00 --------------------------------------------------------------------------------- Stratified by study • Note: hazard ratios for dichotomous • birth weight at 4,000 grams are very • similar between the 2 regression • Models. • In the final models we used x and x2 • since that is the correct scale Cox regression using gestational age and linear paternal age in order to compare to the model with gestational age and quadratic paternal age.

16. Multivariable Confounding Model coefficients and percent change (in red) from model with all 7 variables (blue) for cancer and birth weight dichotomized at 4,000 kg. Already removed from the model are maternal smoking and total pregnancy weight change. • Starting with all confounders in the model, we remove covariates one at a time if they are no longer confounding the relationship birth weight or the other covariates in the model. • This procedure is repeated until a confounder can no longer be dropped as it changes the other coefficients in the model by more than 15% in either direction. • This method leaves a parsimonious multivariable model. We used the MI dataset to check for the multivariable confounding.