Analyzing Unevenly Spaced Longitudinal Count Data

Abstract

In a longitudinal setup, as opposed to equi-spaced count responses, there are situations where an individual patient may provide successive count responses at unevenly spaced time intervals. These unevenly spaced count responses are in general accompanied with covariates information collected at the response occurring time points. Here, the responses and covariates are complete as opposed to certain longitudinal data subject to non-response or missing. The regression analysis of this type of unevenly spaced longitudinal count data is not adequately discussed in the literature. In this paper we propose a dynamic model for unevenly spaced longitudinal Poisson counts and demonstrate the computation of correlations among such count responses through an example with T = 4 time intervals such as 4 weeks as the duration of the longitudinal study. Here, if an individual patient reports a problem (in terms of counts) say at time intervals 1, 3, and 4 (i.e., in first, third and fourth weeks); then 3 count responses collected at these 3 times/weeks would be unevenly spaced. Clearly, this individual had nothing to report at time point 2, i.e., in second week, and hence these 3 responses are considered to be complete. Here, we emphasize that this ‘no response’ in the second week for the individual, is, neither a missing response (or so-called non-response) nor can it be quantified as a zero count because no probability can be assigned for a non-existing event. As far as the total number of time intervals is concerned it can be large but it is usually small in a longitudinal setup. However, for accuracy of correlations, one can make each interval small leading to a large value of T. For inferences, the regression parameters are estimated by using the well known GQL (generalized quasi-likelihood) approach. For the estimation of the unevenly spaced pair-wise correlation index parameters we use a standardized method of moments. The performance of the proposed estimation approaches are examined through an intensive simulation study. The results of this paper should be useful to bio-medical practitioners either currently dealing with this type of unevenly spaced count data or planning for data collection on a similar study.

Appendix: Moment Equations for Pair-Wise Correlation Index Parameters

Table 7 Simulated means (SM) and standard errors (SSE) of parameters of the dynamic model for unevenly spaced count data under design D1 with K = 400, $\sigma _{\gamma }^{2} = 0$

Moment Estimating Equation for ρ ₁₃

Based on the same approach used for ρ₁₂ (see Eq. 4.1)), the moment equation for ρ₁₃ may be constructed as

$$ f_{s,13}-f_s(\rho_{13})=0, $$

(5.1)

where

$$ \begin{array}{@{}rcl@{}} f_{s,13}&=&\sum\limits^{K_{21}+K_{22}+K_{23}}_{i_2=K_{21}+K_{22}+1} \left[\left\{\left( \frac{y_{i_2t_1}-\mu_{i_2t_1}}{\sigma_{i_2t_1}}\right) \left( \frac{y_{i_2t_{3}} -\mu_{i_2t_{3}}}{\sigma_{i_2t_3}}\right)\right\} \right.\\ &&+ \left.\left\{\left( \frac{y_{i_2t_1}-\mu_{i_2t_1}}{\sigma_{i_2t_1}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\}\right]\\ &&+\sum\limits^{K_{31}+K_{32}}_{i_3=K_{31}+1}\left\{\left( \frac{y_{i_3t_1}-\mu_{i_3t_1}} {\sigma_{i_3t_1}}\right)\left( \frac{y_{i_3t_{3}} -\mu_{i_3t_{3}}}{\sigma_{i_3t_3}}\right)\right\}, \end{array} $$

(5.2)

and

$$ \begin{array}{@{}rcl@{}} f_s(\rho_{13})&=&\sum\limits^{K_{21}+K_{22}+K_{23}}_{i_2=K_{21}+K_{22}+1} \left\{\frac{1}{\sigma_{i_2t_1}}\mu_{i_2t_1}\rho_{13}\left( \frac{1}{\sigma_{i_2t_3}} +\frac{1}{\sigma_{i_2t_4}}\rho_{34}\right)\right\} \\ &&+\sum\limits^{K_{31}+K_{32}}_{i_3=K_{31}+1}\left\{\frac{1}{\sigma_{i_3t_1}}\mu_{i_3t_1} \frac{1}{\sigma_{i_3t_3}}\rho_{13}\right\}. \end{array} $$

(5.3)

Moment Estimating Equation for ρ ₁₄

Similar to Eq. 5.1, the moment equation for ρ₁₄ is given by

$$ f_{s,14}-f_s(\rho_{14})=0 $$

(5.4)

where

$$ \begin{array}{@{}rcl@{}} f_{s,14}&=&\sum\limits^{K_{31}+K_{32}+K_{33}}_{i_3=K_{31}+K_{32}+1} \left\{\left( \frac{y_{i_3t_1}-\mu_{i_3t_1}}{\sigma_{i_3t_1}}\right) \left( \frac{y_{i_3t_{4}} -\mu_{i_3t_{4}}}{\sigma_{i_3t_4}}\right)\right\} \end{array} $$

(5.5)

and the corresponding expected function f_s(ρ₁₄) has the formula

$$ \begin{array}{@{}rcl@{}} f_s(\rho_{14})&=&\sum\limits^{K_{31}+K_{32}+K_{33}}_{i_3=K_{31}+K_{32}+1} \left\{\frac{1}{\sigma_{i_3t_1}}\mu_{i_3t_1}\frac{1}{\sigma_{i_3t_4}}\rho_{14}\right\}. \end{array} $$

(5.6)

Moment Estimating Equation for ρ ₂₃

In this case,

$$ f_{s,23}-f_s(\rho_{23})=0, $$

(5.7)

where

$$ \begin{array}{@{}rcl@{}} f_{s,23}&=& \sum\limits^{K_1}_{i_1=1}\sum\limits^1_{d=0}\left[ \left\{\left( \frac{y_{i_11t_1}-\mu_{i_1t_1}}{\sigma_{i_1t_1}}\right) \left( \frac{y_{i_1t_{3+d}} -\mu_{i_1t_{3+d}}}{\sigma_{i_1t_{3+d}}}\right)\right\}\right.\\ &&+ \left. \left\{\left( \frac{y_{i_1t_2}-\mu_{i_1t_2}}{\sigma_{i_1t_2}}\right) \left( \frac{y_{i_1t_{3+d}} -\mu_{i_1t_{3+d}}}{\sigma_{i_1t_{3+d}}}\right)\right\}\right] \\ &&+\sum\limits^{K_{21}}_{i_2=1}\left[ \left\{\left( \frac{y_{i_2t_1}-\mu_{i_2t_1}}{\sigma_{i_2t_1}}\right) \left( \frac{y_{i_2t_{3}} -\mu_{i_2t_{3}}}{\sigma_{i_2t_3}}\right)\right\}\right.\\ &&+ \left. \left\{\left( \frac{y_{i_2t_2}-\mu_{i_2t_2}}{\sigma_{i_2t_2}}\right) \left( \frac{y_{i_2t_{3}} -\mu_{i_2t_{3}}}{\sigma_{i_2t_3}}\right)\right\}\right] \\ &&+\sum\limits^{K_2}_{i_2=K_{21}+K_{22}+K_{23}+1}\sum\limits^1_{d=0} \left\{\left( \frac{y_{i_2t_2}-\mu_{i_2t_2}}{\sigma_{i_2t_2}}\right) \left( \frac{y_{i_2t_{3+d}} -\mu_{i_2t_{3+d}}}{\sigma_{i_2t_{3+d}}}\right)\right\} \\ &&+{\sum}^{K_{31}+K_{32}+K_{33}+K_{34}}_{i_3=K_{31}+K_{32}+K_{33}+1} \left\{\left( \frac{y_{i_2t_2}-\mu_{i_2t_2}}{\sigma_{i_2t_2}}\right) \left( \frac{y_{i_2t_{3}} -\mu_{i_2t_{3}}}{\sigma_{i_2t_3}}\right)\right\}, \end{array} $$

(5.8)

and the corresponding expected function is given by

$$ \begin{array}{@{}rcl@{}} f_s(\rho_{23})&=&{\sum}^{K_1}_{i_1=1}\left[\left\{\frac{1}{\sigma_{i_1t_1}}\mu_{i_1t_1} \rho_{23}\rho_{12}\left( \frac{1}{\sigma_{i_1t_3}}+\frac{1}{\sigma_{i_1t_4}}\rho_{34}\right)\right\} \right.\\ &&+\left. \left\{\frac{1}{\sigma_{i_1t_2}}\mu_{i_1t_2}\rho_{23}\left( \frac{1}{\sigma_{i_1t_3}}+ \frac{1}{\sigma_{i_1t_4}}\rho_{34}\right)\right\} \right]\\ &&+{\sum}^{K_{21}}_{i_2=1}\left[\left\{\frac{1}{\sigma_{i_2t_2}}\mu_{i_2t_2} \frac{1}{\sigma_{i_2t_3}}\rho_{23}\right\} +\left\{\frac{1}{\sigma_{i_2t_1}}\mu_{i_2t_1}\frac{1}{\sigma_{i_2t_3}} \rho_{23}\rho_{12}\right\}\right] \\ &&+{\sum}^{K_2}_{i_2=K_{21}+K_{22}+K_{23}+1}\left\{ \frac{1}{\sigma_{i_2t_2}}\mu_{i_2t_2}\rho_{23}\left( \frac{1}{\sigma_{i_2t_3}} +\frac{1}{\sigma_{i_2t_4}}\rho_{34}\right)\right\}\\ &&+{\sum}^{K_{31}+K_{32}+K_{33}+K_{34}}_{i_3=K_{31}+K_{32}+K_{33}+1} \left\{\frac{1}{\sigma_{i_3t_2}}\mu_{i_3t_2}\frac{1}{\sigma_{i_3t_3}} \rho_{23}\right\}. \end{array} $$

(5.9)

Moment Estimating Equation for ρ ₂₄

This moment equation is written as

$$ f_{s,24}-f_s(\rho_{24})=0, $$

(5.10)

where

$$ \begin{array}{@{}rcl@{}} f_{s,24}&=&{\sum}^{K_{21}+K_{22}}_{i_2=K_{21}+1} \left[ \left\{\left( \frac{y_{i_2t_1}-\mu_{i_2t_1}}{\sigma_{i_2t_1}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\}\right.\\ &&+ \left. \left\{\left( \frac{y_{i_2t_2}-\mu_{i_2t_2}}{\sigma_{i_2t_2}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\} \right]\\ &&+{\sum}^{K_{31}+K_{32}+K_{33}+K_{34}+K_{35}}_{i_3=K_{31}+K_{32}+K_{34}+1} \left[\left\{\left( \frac{y_{i_3t_2}-\mu_{i_3t_2}}{\sigma_{i_3t_2}}\right) \left( \frac{y_{i_3t_{4}} -\mu_{i_3t_{4}}}{\sigma_{i_3t_4}}\right)\right\}\right],\\ \end{array} $$

(5.11)

and

$$ \begin{array}{@{}rcl@{}} f_s(\rho_{24})&=&{\sum}^{K_{21}+K_{22}}_{i_2=K_{21}+1} \left[\left\{\frac{1}{\sigma_{i_2t_2}}\mu_{i_2t_2}\frac{1}{\sigma_{i_2t_4}}\rho_{24}\right\} +\left\{\frac{1}{\sigma_{i_2t_1}}\mu_{i_2t_1}\frac{1}{\sigma_{i_2t_4}}\rho_{24}\rho_{12}\right\} \right]\\ &&+{\sum}^{K_{31}+K_{32}+K_{33}+K_{34}+K_{35}}_{i_3=K_{31}+K_{32}+K_{34}+1} \left\{\frac{1}{\sigma_{i_3t_2}}\mu_{i_3t_2}\frac{1}{\sigma_{i_3t_4}}\rho_{24}\right\}. \end{array} $$

(5.12)

Moment Estimating Equation for ρ ₃₄

The moment equation for the last correlation index parameter ρ₃₄ is written as

$$ f_{s,34}-f_s(\rho_{34})=0, $$

(5.13)

where

$$ \begin{array}{@{}rcl@{}} f_{s,34}&=&{\sum}^{K_1}_{i_1=1}{\sum}^{T-1}_{u=1} \left\{\left( \frac{y_{i_1t_u}-\mu_{i_1t_u}}{\sigma_{i_1t_u}}\right) \left( \frac{y_{i_1t_{4}} -\mu_{i_1t_{4}}}{\sigma_{i_1t_4}}\right)\right\}\\ &&+{\sum}^{K_{21}+K_{22}+K_{23}}_{i_2=K_{21}+K_{22}+1} \left[\left\{\left( \frac{y_{i_2t_1}-\mu_{i_2t_1}}{\sigma_{i_2t_1}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\} \right.\\ &&+ \left. \left\{\left( \frac{y_{i_2t_3}-\mu_{i_2t_3}}{\sigma_{i_2t_3}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\}\right]\\ &&+{\sum}^{K_2}_{i_2=K_{21}+K_{22}+K_{23}+1} \left[\left\{\left( \frac{y_{i_2t_2}-\mu_{i_2t_2}}{\sigma_{i_2t_2}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\}\right.\\ &&+\left. \left\{\left( \frac{y_{i_2t_3}-\mu_{i_2t_3}}{\sigma_{i_2t_3}}\right) \left( \frac{y_{i_2t_{4}} -\mu_{i_2t_{4}}}{\sigma_{i_2t_4}}\right)\right\}\right]\\ &&+{\sum}^{K_3}_{i_3=K_{31}+K_{32}+K_{33}+K_{34}+K_{35}+1} \left\{\left( \frac{y_{i_3t_3}-\mu_{i_3t_3}}{\sigma_{i_3t_3}}\right) \left( \frac{y_{i_3t_{4}} -\mu_{i_3t_{4}}}{\sigma_{i_3t_4}}\right)\right\},\\ \end{array} $$

(5.14)

and f_s(ρ₃₄) has the formula

$$ \begin{array}{@{}rcl@{}} f_s(\rho_{34})&=& {\sum}^{K_1}_{i_1=1}\left\{\frac{1}{\sigma_{i_1t_4}}\rho_{34} \left( \frac{1}{\sigma_{i_1t_1}}\mu_{i_1t_1}\rho_{12}\rho_{23} +\frac{1}{\sigma_{i_1t_2}}\mu_{i_1t_2}\rho_{23}+\frac{1}{\sigma_{i_1t_3}} \mu_{i_1t_3}\right)\right\}\\ &&+{\sum}^{K_{21}+K_{22}+K_{23}}_{i_2=K_{21}+K_{22}+1} \left\{\frac{1}{\sigma_{i_2t_4}}\rho_{34} \left( \frac{1}{\sigma_{i_2t_1}}\mu_{i_2t_1}\rho_{13}+\frac{1}{\sigma_{i_2t_3}} \mu_{i_2t_3}\right)\right\} \\ &&+{\sum}^{K_2}_{i_2=K_{21}+K_{22}+K_{23}+1} \left\{\frac{1}{\sigma_{i_2t_4}}\rho_{34}\left( \frac{1}{\sigma_{i_2t_2}}\mu_{2_1t_2}\rho_{23} +\frac{1}{\sigma_{i_2t_3}}\mu_{i_2t_3}\right)\right\} \\ &&+ {\sum}^{K_3}_{i_3=K_{31}+K_{32}+K_{33}+K_{34}+K_{35}+1} \left[\frac{1}{\sigma_{i_3t_3}}\mu_{i_3t_3}\frac{1}{\sigma_{i_3t_4}}\rho_{34}\right]. \end{array} $$

(5.15)