Substantive analyses for Aim 3

3. Materials and Methods

3.8 Statistical analyses

3.8.6 Substantive analyses for Aim 3

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Covariates Functional form Degrees of freedom

Model 1 Deprivation index Restricted cubic splines (nk = 4) 3

Age Restricted cubic splines (nk = 4) 3

Race and sex Race (whites vs blacks); Sex 2

Model 2 Model 1 variables

Income (<$15000, $15000-24999 & ≥$25000) 2 Education < HS, HS/vocational training/junior

college, college degree or higher)

2 Model 3 Model 2 variables

Smoking Status never/former/current <19.5 pack-

years/current ≥ 19.5 pack-years 3

Alcohol intake Linear 1

BMI Restricted cubic splines (nk = 4) 3

History of diabetes, hypertension, high

cholesterol, MI/CABG, stroke

All yes/no 5

Physical activity in met-hrs

Linear + quadratic 2

Total 26

We would equally present the ICC (computed using the latent variable approach) as it is provides information about the proportion of the variance that is explained by differences across neighborhoods which has a greater public health relevance. Hazard ratios for 1 interquartile range increase in deprivation index would equally be presented as these have an intuitive interpretation: i.e. the hazard of the event occurring for a typical person in the middle of the upper half of the distribution to the hazard of the event for a typical person in the middle of the lower half of the distribution.

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𝑦 = 𝑐 + log 𝑊 𝐻⁄ ^𝑛, assuming a slope of 1.

𝑦 = 𝑐 + log 𝑊 + log 𝐻^−𝑛 𝑦 = 𝑐 + log 𝑊 − nlog 𝐻

So the absolute value of the ratio of the coefficients of log W and log H is n.

In the case specific case of the Cox Model for the log hazard of HF (with no intercept), when log hazard of HF is regressed on log W and log H we get

𝑦 = 𝛼₁log 𝑊 + 𝛼₂log 𝐻

Where y = log hazard (HF), 𝛼₂ is negative if y and W/Hⁿ are positively correlated and vice versa.

And n would be given by |𝛼₂⁄𝛼₁|

Second, in separate Cox models, we would regress the restricted cubic splines of the natural log of the data-derived weight-height index and that of BMI on the log hazard of HF. Then, model fit statistics (LR chi square, χ² and AIC) would be used to compare the performance of the data-derived weight-index versus that of BMI in relation to a model utilizing restricted cubic splines of log weight and log height.

Third, we would run models with log BMI and log height to see if log height is still significant in a model containing BMI. We would also compare the effect size for a 1 interquartile range increase in BMI and the computed W/Hⁿ index.

Fourth, we would use multivariable Cox models which take into account nonlinearity and non- additivity to model a flexible dose-response association between the better performing weight-height index (W/Hⁿ) (modelled using restricted cubic splines with 5 evenly spaced knots) and HF risk adjusting for relevant covariates in a sequential fashion as shown in the table below:

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Covariates Functional form Degrees of

freedom Model 1 Weight-height index Restricted cubic splines (nk = 5) 4

Age Restricted cubic splines (nk = 4) 3

Race and sex Race (whites vs blacks); Sex (women vs Women)

2 W/Hⁿ×race terms Linear + non-linear interaction

terms.

4 Model 2 Model 1 variables

Income (<$15000, $15000-24999 &

≥$25000) 2

Education < HS, HS/vocational training/junior college, college degree or higher)

2 Smoking Status never/former/current <19.5 pack-

years/current ≥ 19.5 pack-years 3

Alcohol intake Linear 1

Physical activity in MET-hrs Linear + quadratic 2

Model 3 Model 3 variables History of diabetes, hypertension, high cholesterol,

MI/CABG, stroke

All yes/no 5

Total 28

Interactions between W/Hⁿ and race as well as sex would be tested. Fourth, we would repeat the multivariable models for the relationship between waist circumference and HF. We would present plots of predicted probabilities (or HRs) of incident heart failure versus weight-height index stratified by race and/or sex. These analyses would be repeated using waist circumference.

For the analyses for WC, we have data for 3304 participants and there are 251 cases observed among these participants. Using the rule of thumb of 10-15 cases per df (or parameter to be estimated) that leaves us with 17-25 degrees of freedom allowed in our model. We would reduce the df spent on some less important covariates (based on prior literature). Below is the proposed df to be spent in the multivariable cox model for WC. Formal power calculations for WC are presented in section 4.9.

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Covariates Functional form Degrees of freedom

Model 1 WC Restricted cubic splines (nk = 4) 3

Age Restricted cubic splines (nk = 4) 3

Race and sex Race (whites vs blacks); Sex 2

Model 2 Model 1 variables

Income (<$15000, ≥$15000) 1

Education < HS, HS/vocational training/junior college, college degree or higher)

2

Smoking Status Never, former and current 2

Alcohol intake Linear 1

Physical activity in MET-hrs Linear + quadratic 2

Model 4 Model 3 variables History of diabetes,

hypertension, high cholesterol, MI/CABG,

stroke

All yes/no 5

Total 21

For the association between (W/Hⁿ) and post-HF survival we would perform similar analyses as we did for the association with HF risk. Hazard ratios for 1 interquartile range increase in W/Hⁿ, WC and BMI would equally be presented as these have an intuitive interpretation: i.e. the hazard of the event occurring for a typical person in the middle of the upper half of the distribution to the hazard of the event for a typical person in the middle of the lower half of the distribution.

For all our models we would verify the PHM assumption by utilizing Schoenfeld residuals from the Cox Models and log (-log) plots. Martingale residuals and dfbetas would be used to investigate the functional form of predictor variables and influential observations respectively.

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