The novel Coronavirus designated SARS-CoV-2 appeared in December 2019 to initiate a pandemic of respiratory illness known as COVID-19, which had been an unprecedented global public health crisis, and the safe and effective COVID-19 vaccinations are vital for the global strategy to combat the pandemic. However, the population must reach a sufficient vaccination rate, i.e. 60-70%, to achieve herd immunity. Vaccine hesitancy, or "delay in acceptance or refusal of vaccination despite availability of vaccination services", could affect vaccination rate and the ability to establish herd immunity. Factors that affect the attitude towards acceptance of vaccination include: complacency (do not perceive a need for a vaccine), convenience (access) and confidence (do not trust vaccine or provider). Therefore, determining the factors associated with COVID-19 vaccine hesitancy are important for public health developing strategies targeting the voluntarily vaccine-hesitant individuals. To evaluate the vaccination compliance rates of individuals living in the U.S., a longitudinal study was used to examine the individual's attitudes toward vaccines over a six-month period. Beginning in March 2020 (early phase of the pandemic), researchers collects attitudes from a cohort of the same participants, N = 407, every month. The primary outcome was the COVID-19 Vaccine Hesitancy Score (vhs19), which can range between 0 and 100, with higher scores indicating lower COVID-19 vaccine hesitancy, i.e. more positive attitude toward vaccination. Additional available data included ID = Participant ID number. month = Time of data collection as follows gender Gender indicator, e.g. 0=female, 1=male. age Age of participant at the baseline, i.e. month = 1. SES = Socioeconomic status, e.g. 0=low SES, 1-high SES. political Participant's political party affiliation, e.g. 0-Democratic, 1-Republican. = = month= 1(March 2020), 2(April 2020),..., 6(August 2020). = The primary goal of the analysis is to investigate whether or not evidence for vaccine attitudes (vhs 19) exists with respect to political party affiliation (political) during an unprecedented public health crisis. Figure 1 in the Appendix A reports a plot with vaccination attitudes by political party affiliation for each participant profiles (March-August 2020). Note that we may only focus on a few of the available covariates in the dataset for this problem. 4. (10 points) The remainder of the question concerns some plausible extensions of modeling effort. You can choose an appropriate model to answer this question. (a) (5 points) It is hypothesized that the rate of change in vhs 19 is the same by political group for the first two months, then differs by group after that. Write down an appropriate model to address this scientific hypothesis (denote your model parameters by ao, a1,..., ., ap). (b) (5 points) You may ignore part (a) from here on. Now reviewers of the research ask the research to determine if there is heterogeneity in the rate of change of vhs 19 among partic- ipants (conditional upon political affiliation). Can your chosen model address reviewers' question? If yes, state explicitly how you would go about testing whether or not there was heterogeneity in the rate of change of vhs 19 among participants. If not, explain your reasoning. #### Model modA > osfcovid19. Aug = subset (osfcovid19, month == 6) > modA = > summary (modA) 1m (formula = vhs19 political age, data = 1m (vhs 19 Coefficients: political age, data osfcovid19. Aug) osfcovid19. Aug) Estimate Std. Error (Intercept) 66.340944 0.370043 political -10.790488 0.208328 age 0.020563 0.009659 age Appendix B Signif. codes: 0 ***' 0.001 '**' 0.01 * 0.05 0.1 1 Residual standard error: 2.098 on 404 degrees of freedom Multiple R-squared: 0.8691, Adjusted R-squared: 0.8685 F-statistic: 1342 on 2 and 404 DF, p-value: < 2.2e-16 > confint (modA) 2.5 % 97.5 % (Intercept) 65.613494478 67.06839402 political -11.200030614 -10.38094612 0.001574553 0.03955191 Estimate (Intercept) 62.1704 gender -1.0731 > modA2 = 1m (political > summary (modA2) Coefficients: #### Confounder: gender > modA1 = 1m (vhs 19 gender, data = osfcovid19. Aug) > summary (modA1) Coefficients: Estimate (Intercept) 0.45631 gender 0.09593 t value. 179.279 -51.796 2.129 Std. Error 0.4019 0.5719 Std. Error 0.03476 0.04946 Pr (>|t|) summary (modB) Generalized least squares fit by maximum likelihood Model: vhs19 age + month political Data: osfcovid19 AIC BIC logLik 10558.67 10599.28 -5272.336 Coefficients: (Intercept) age month political month: political = #### Model modC > modC gls (vhs 19 + data-osfcovid19, correlation Coefficients: age (Intercept) age month political = Appendix C SES gender month: political month: SES month: gender month political, data=osfcovid19, corCompSymm (form = 1 | ID), method = 'ML') Value Std. Error t-value p-value 67.38018 0.20069079 335.7413 0.01896 0.00415485 4.5638 -0.18218 0.03492432 -5.2164 -7.95862 0.19415006 -40.9921 -0.47736 0.04920941 -9.7006 Model: vhs 19 Data: osfcovid19 AIC BIC logLik 10551.71 10615.51 -5264.854 + > summary (modC) Generalized least squares fit by maximum likelihood 0 0 age month political + month*SES + month*gender, correlation = corCompSymm (form = 1 | ID), method = 'ML') 0 0 0 age + month political + month * SES + month * gender Value Std. Error t-value p-value 67.45976 0.23014435 293.11934 0.0000 0.01812 0.00410459 4.41389 0.0000 -0.21565 0.04520894 -4.77000 0.0000 -7.92564 0.19451912 -40.74479 0.0000 0.23666 0.20825625 1.13641 0.2559 -0.28558 0.19476820 -1.46626 0.1427 -0.48213 0.04946676 -9.74648 0.0000 0.02717 0.05297598 0.51284 0.6081 0.05519 0.04951963 1.11459 0.2651