Estimate the population variation in annual growth rates of Nepali children and explore whether this differs by mothers parity

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Biostatistics 140.653

Third Term, 2022

Problem Set 3 

Instructions: Feel free to discuss and complete the analysis with other students. However, each student must write-up their own solutions. Write as if for a scientific journal. Be brief and accurate. Submit your text answers along with your code in an html or pdf file generated via RMarkdown.

Due in CoursePlus drop box: Friday, March 11 by 12:00pm (noon) EST

For this problem set, use the complete Nepal Anthropometry Study (NAS) Dataset with up to 5 measurements on each child over time.

The goals of the analysis are to:

1) Determine if the average growth rates of children differ by mother’s parity (number of previous live births)

2) Estimate the population variation in annual growth rates of Nepali children and explore whether this differs by mother’s parity

Part I: Get familiar with the data

1. Make a table of mother’s parity (alive variable). Ideally, we would compare children of nulliparous women to categories of women of parity > 0. However, in this dataset, there are only 19 children from nulliparous women. So, we will create two categories of women: parity ≤ 3 (i.e. 1 to 4 live births) vs. parity > 3 (5 or more live births).

2. Make a spaghetti plot of children’s weight as a function of age; connecting the measured weights within a child over time. Color code the data by parity group. Add smoothing splines for each parity group. Note any similarities or differences in the growth rates across the two parity groups.

Part II: Model checking and recommendations

Fit the following model to the data:

𝑌𝑌𝑖𝑖𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 + 𝛽𝛽2𝐼𝐼(𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑖𝑖 > 3) + 𝛽𝛽3𝐼𝐼(𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑖𝑖 > 3)𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 + 𝜀𝜀𝑖𝑖𝑖𝑖, 𝜀𝜀𝑖𝑖𝑖𝑖~𝑁𝑁(0, 𝜎𝜎2), 𝐶𝐶𝐶𝐶𝐶𝐶�𝜀𝜀𝑖𝑖𝑖𝑖, 𝜀𝜀𝑖𝑖𝑖𝑖� = 0,

where i indicates the child (i =1, …, 200) and j denotes the follow-up (j = 1, 2, 3, 4, 5).

1. Conduct appropriate checking of this model; i.e. check for appropriateness of the mean model, and the independence and constant variance assumptions for the residuals.

2. Based on your model checking, propose an alternative model for the data that can address the first goal of the analysis (i.e. determine if the growth rates of children differ by mother’s parity (number of previous live births) while satisfying the observed patterns in data with respect to the mean model and distribution of residuals. NOTE: If you modify the mean model, you may want to iterate between model checking for the mean.

Part III: Marginal model for longitudinal data

1. Use the gls function in R to fit the model you proposed in Part I. From the fit of the model, compute the estimated 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(𝜀𝜀𝑖𝑖0, 𝜀𝜀𝑖𝑖𝑖𝑖) for j = 1, 2, 3, 4 where the follow-up visits (fuvisit) have values 0 (baseline) and 1, 2, 3, 4 (representing the 4 follow-up visits each 4 months apart).

2. Conduct a likelihood ratio test to address the first goal of the analysis; i.e. to determine if the average growth rates of children differ by mother’s parity (number of previous live births).

3. Fit the mean model you proposed in Part I using the gee function but where you allow the correlation structure to be “independence”. The gee function will produce standard error estimates assuming the independence assumption (labeled as “naïve” or “model-based” standard error estimates) and “robust” standard error estimates (using the Huber-White sandwich estimator). Compare the estimated coefficients and standard errors from the gls and gee model fits

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