Prof Dr Slavica Ražić 2 Prof Dr Ari Ivaska

A model for longitudinal density was developed by fitting a linear mixed model with BI-RADS density as the outcome (treated as an integer to approximate linear relationships with density). This model uses a similar two-level hierarchical structure as that in Chapter 2, whereby the base level is each density measure at each time point, and the second level is each woman (2.2.4.4).

The linear mixed model for this study is described below. Breast density 𝑦_𝑖𝑗 for woman 𝑖 = 1, … , 𝑛 at time 𝑗 = 1, …, m_𝑖 was modelled as:

𝑦_𝑖𝑗= 𝛽₀+ 𝑢_0𝑖+ (𝛽₁+ 𝑢_1𝑖)𝑎𝑔𝑒_𝑖𝑗+ 𝛽₂𝑎𝑔𝑒_𝑖𝑗2 + 𝛽₃𝑎𝑔𝑒_𝑖𝑗3+ 𝛽₄𝑎𝑔𝑒_𝑖𝑗4+ 𝛽₅𝐵𝑀𝐼_𝑖𝑗 + 𝛽₆𝑎𝑔𝑒_𝑖𝑗𝐵𝑀𝐼_𝑖𝑗+ 𝑒_𝑖𝑗 ;

where 𝛽₀ is an overall intercept, 𝑎𝑔𝑒_𝑖𝑗 is the age for woman 𝑖 at time 𝑗, 𝛽₁ is the slope for age, 𝛽2 is the slope for age-squared, 𝛽3 is the slope for age-cubed, 𝛽4 is the slope for age to the

power of four, 𝐵𝑀𝐼_𝑖𝑗 is the BMI for woman 𝑖 at time 𝑗, 𝛽₅ is the slope for BMI, 𝛽₆ is the interaction effect for BMI and the linear age term, and 𝑒_𝑖𝑗 is a random error. The term that allowed for differences between-women in their overall density level is the independent random intercept 𝑢_0𝑖 for woman 𝑖. The term that allowed for differences between-women in their age slope is the independent random slope 𝑢_1𝑖 for woman 𝑖. In other words, the random age slopes allowed each woman to have density trajectories that deviated from the average trajectory through time. Age had a non-linear relationship with density, as has been seen previously (141, 255, 317). The model is completed by assuming normal distributions for 𝒖_𝑖= (𝑢_0i,𝑢_1i) and 𝑒_𝑖𝑗, with zero mean, unknown variances and: zero covariance between 𝑒_𝑖𝑗 of the same woman or different women, zero covariance between 𝒖_𝑖 and 𝑒_𝑖𝑗 of the same woman or different women, zero covariance between 𝒖_𝑖 of different women, and unknown covariance between 𝑢_0𝑖 and 𝑢_1𝑖 of the same woman. The model was fitted by maximum likelihood (2.2.4.7). To test 𝛽𝑘 = 0 for 𝑘 = 0, …,6, Wald tests were applied (2.2.4.9).

The linear mixed model building strategy was based on a series of likelihood ratio tests to assess goodness of fit with various polynomial terms and interactions as well as visual assessment of

87 graphs plotting predicted density against age and BMI. Standard errors for the longitudinal model were calculated using robust sandwich estimators (318-320). These were calculated empirically without making any assumptions on the structure of heteroscedasticity (unequal variance across variable values) in the model.

From this linear mixed model, each woman’s random effects, 𝒖_𝑖= (𝑢_0i, 𝑢_1i), were then predicted using Empirical Bayes, as described below (287).

To better understand the Empirical Bayes prediction, it is useful to describe a linear mixed model in its matrix form, whereby, for woman 𝑖 = 1, … , 𝑛 with 𝑗 = 1, …, m_𝑖 time points:

𝒚_𝑖 = 𝑿_𝑖𝜷 + 𝒁_𝑖𝒖_𝑖+ 𝒆_𝑖 ; where:

 𝒚_𝑖 = (𝑦_𝑖1,… , 𝑦_𝑖m𝑖)𝑇is the m_𝑖 x 1 column vector of observed outcomes for woman 𝑖  𝑿𝑖 = (

𝑥_𝑖11 … 𝑥_𝑖1𝑎

⋮ ⋱ ⋮

𝑥𝑖m𝑖1 … 𝑥𝑖m𝑖𝑎

) is the m𝑖 x 𝑎 design matrix of observed predictors for fixed

effects 𝑝 = 1, …, 𝑎 for woman 𝑖

 𝜷 = (𝛽0, …, 𝛽a−1)𝑇 is the 𝑎 x 1 column vector of regression coefficients for fixed effects

𝑝 = 1, … , 𝑎  𝒁𝑖 = (

𝑥_𝑖11 … 𝑥_𝑖1𝑏

⋮ ⋱ ⋮

𝑥𝑖m𝑖1 … 𝑥𝑖m𝑖𝑏

) is the m𝑖 x 𝑏 design matrix of observed predictors for random

effects 𝑞 = 1, …, 𝑏 for woman 𝑖

 𝒖𝑖= (𝑢0i, …, 𝑢b−1 i)𝑇 is the 𝑏 x 1 column vector of unobserved random effects 𝑞 = 1, … , 𝑏

for woman 𝑖  𝒆𝑖 = (𝑒𝑖1, …, 𝑒𝑖m𝑖)

𝑇

is the m𝑖 x 1 column vector of unobserved random errors for woman 𝑖

Under the above model, density measures for woman 𝑖 have mean = 𝑿_𝑖𝜷 + 𝒁_𝑖𝒖_𝑖 and variance = 𝑽𝑖= 𝒁𝑖𝚺𝒁𝑖𝑇+ σ2𝑰𝑚𝑖 .

Recalling from Chapter 2, 𝚺 is the variance-covariance matrix, and σ2𝑰_𝑚𝑖=𝑬_𝑖 which is the variance of the residuals for woman 𝑖, with σ2 being the sample residual variance and 𝑰_𝑚𝑖 being the 𝑚_𝑖 x 𝑚_𝑖 identity matrix. The values for 𝒚_𝑖, 𝑿_𝑖 and 𝒁_𝑖 are measured, and estimates of parameters 𝜷, 𝚺 and σ2 _{are obtained by generalised least squares, which corresponds to} maximum likelihood assuming normality of 𝒖_𝑖 and 𝒆_𝑖 (2.2.4.4).

88 Empirical Bayes can be used to predict the random effects, 𝒖_𝑖, by entering the observed values and estimated parameters into the following equation:

𝔼 (𝒖_𝑖 | 𝑿_𝑖,𝒁_𝑖, 𝜷̂,𝚺̂,σ̂2_{) =𝚺̂𝒁} 𝑖 𝑇_𝑽̂ 𝑖 −1 _(𝒚 𝑖− 𝑿𝑖𝜷̂)

In this study, the above equation was used to predict the random intercept and random slope for each woman 𝑖 = 1, … , 𝑛 at each time point 𝑗 = 1, … , m_𝑖 using data from her most recent and previous observations only. Therefore, each woman’s individual observations had a uniquely determined predicted random intercept and random slope.

A predicted density value was then calculated by entering the observed values, estimated parameters and predicted random effects into the equation:

𝔼 (𝒚_𝑖 | 𝑿_𝑖, 𝒁_𝑖, 𝜷̂,𝚺̂,σ̂2_{) = 𝑿}

𝑖𝜷̂ + 𝒁𝑖𝒖̂𝑖

In the analysis, these Bayes predicted density measures are referred to as the longitudinal density measures.

At this point, each observation for each woman had a corresponding baseline density value (the starting value for woman 𝑖), most recent density value (the updated density value for woman 𝑖 at time 𝑗), longitudinal density value (the updated Bayes predicted density for woman 𝑖 at time 𝑗), age at baseline (the baseline age for woman 𝑖), baseline BMI (the baseline BMI for woman 𝑖), and most recent BMI (the updated BMI for woman 𝑖 at time 𝑗).