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A probit- log- skew-normal mixture model for repeated measures data with excess zeros, with application to a cohort study of paediatric respiratory symptoms.

Mahmud S, Lou WW, Johnston NW - BMC Med Res Methodol (2010)

Bottom Line: The severity of the symptoms was also greater for the week a child was viral positive.For DAVIS data the random effects probit -log skew normal model fits significantly better than the random effects probit -log normal model, endorsing our parametric choice for the model.The simulation study indicates that our proposed model seems to be robust to misspecification of the distribution of the positive skewed response.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Community Health Sciences, The Aga Khan University, Stadium Road, PO Box 3500, Karachi 74800 Pakistan. sadia.mahmud@aku.edu

ABSTRACT

Background: A zero-inflated continuous outcome is characterized by occurrence of "excess" zeros that more than a single distribution can explain, with the positive observations forming a skewed distribution. Mixture models are employed for regression analysis of zero-inflated data. Moreover, for repeated measures zero-inflated data the clustering structure should also be modeled for an adequate analysis.

Methods: Diary of Asthma and Viral Infections Study (DAVIS) was a one year (2004) cohort study conducted at McMaster University to monitor viral infection and respiratory symptoms in children aged 5-11 years with and without asthma. Respiratory symptoms were recorded daily using either an Internet or paper-based diary. Changes in symptoms were assessed by study staff and led to collection of nasal fluid specimens for virological testing. The study objectives included investigating the response of respiratory symptoms to respiratory viral infection in children with and without asthma over a one year period. Due to sparse data daily respiratory symptom scores were aggregated into weekly average scores. More than 70% of the weekly average scores were zero, with the positive scores forming a skewed distribution. We propose a random effects probit/log-skew-normal mixture model to analyze the DAVIS data. The model parameters were estimated using a maximum marginal likelihood approach. A simulation study was conducted to assess the performance of the proposed mixture model if the underlying distribution of the positive response is different from log-skew normal.

Results: Viral infection status was highly significant in both probit and log-skew normal model components respectively. The probability of being symptom free was much lower for the week a child was viral positive relative to the week she/he was viral negative. The severity of the symptoms was also greater for the week a child was viral positive. The probability of being symptom free was smaller for asthmatics relative to non-asthmatics throughout the year, whereas there was no difference in the severity of the symptoms between the two groups.

Conclusions: A positive association was observed between viral infection status and both the probability of experiencing any respiratory symptoms, and their severity during the year. For DAVIS data the random effects probit -log skew normal model fits significantly better than the random effects probit -log normal model, endorsing our parametric choice for the model. The simulation study indicates that our proposed model seems to be robust to misspecification of the distribution of the positive skewed response.

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Log (LRT > 0) versus the week of the year
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Figure 3: Log (LRT > 0) versus the week of the year

Mentions: Covariates associated with the severity of LRT symptoms were subject's age and sex, viral infection status and week of follow-up (log skew normal component). There was no significant difference in the severity of the LRT symptoms between asthmatic and non-asthmatic children (p-value = 0.2758). For the week a child was viral positive the severity of LRT symptoms was significantly greater than for the week he/she was viral negative (p-value < 0.0001, beta = 0.744). There was some marginal evidence that the severity of LRT symptoms was lesser for the male relative to the female children (p-value = 0.0938, beta = -0.1286). As for the probit component, the quadratic term for week was significant in the log-skew normal component. Moreover, there was a significant interaction between the age of a child and (the linear term of) the week of follow (p-value = 0.0069). We also examined the interaction of age with the quadratic term in week (that is age*week*week), however that was insignificant based on the Wald, likelihood, AIC and BIC criteria. In Figure 3 we plot predicted values of log(LRT > 0) from the fitted model versus week of the year for two age groups, < 8 years (mean age) and ≥8 years, along with mean log(LRT > 0) values at each week computed from the data. (In Figure 4 mean LRT > 0 scores on the original scale, computed from the data, are plotted versus week of follow-up). The severity of LRT symptoms was higher in the beginning of the year, decreased in summer and increased again by the end of the year. The severity of LRT symptoms was higher for children younger than 8 years relative to older children, and seemed to exhibit a more pronounced seasonal pattern.


A probit- log- skew-normal mixture model for repeated measures data with excess zeros, with application to a cohort study of paediatric respiratory symptoms.

Mahmud S, Lou WW, Johnston NW - BMC Med Res Methodol (2010)

Log (LRT > 0) versus the week of the year
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2902491&req=5

Figure 3: Log (LRT > 0) versus the week of the year
Mentions: Covariates associated with the severity of LRT symptoms were subject's age and sex, viral infection status and week of follow-up (log skew normal component). There was no significant difference in the severity of the LRT symptoms between asthmatic and non-asthmatic children (p-value = 0.2758). For the week a child was viral positive the severity of LRT symptoms was significantly greater than for the week he/she was viral negative (p-value < 0.0001, beta = 0.744). There was some marginal evidence that the severity of LRT symptoms was lesser for the male relative to the female children (p-value = 0.0938, beta = -0.1286). As for the probit component, the quadratic term for week was significant in the log-skew normal component. Moreover, there was a significant interaction between the age of a child and (the linear term of) the week of follow (p-value = 0.0069). We also examined the interaction of age with the quadratic term in week (that is age*week*week), however that was insignificant based on the Wald, likelihood, AIC and BIC criteria. In Figure 3 we plot predicted values of log(LRT > 0) from the fitted model versus week of the year for two age groups, < 8 years (mean age) and ≥8 years, along with mean log(LRT > 0) values at each week computed from the data. (In Figure 4 mean LRT > 0 scores on the original scale, computed from the data, are plotted versus week of follow-up). The severity of LRT symptoms was higher in the beginning of the year, decreased in summer and increased again by the end of the year. The severity of LRT symptoms was higher for children younger than 8 years relative to older children, and seemed to exhibit a more pronounced seasonal pattern.

Bottom Line: The severity of the symptoms was also greater for the week a child was viral positive.For DAVIS data the random effects probit -log skew normal model fits significantly better than the random effects probit -log normal model, endorsing our parametric choice for the model.The simulation study indicates that our proposed model seems to be robust to misspecification of the distribution of the positive skewed response.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Community Health Sciences, The Aga Khan University, Stadium Road, PO Box 3500, Karachi 74800 Pakistan. sadia.mahmud@aku.edu

ABSTRACT

Background: A zero-inflated continuous outcome is characterized by occurrence of "excess" zeros that more than a single distribution can explain, with the positive observations forming a skewed distribution. Mixture models are employed for regression analysis of zero-inflated data. Moreover, for repeated measures zero-inflated data the clustering structure should also be modeled for an adequate analysis.

Methods: Diary of Asthma and Viral Infections Study (DAVIS) was a one year (2004) cohort study conducted at McMaster University to monitor viral infection and respiratory symptoms in children aged 5-11 years with and without asthma. Respiratory symptoms were recorded daily using either an Internet or paper-based diary. Changes in symptoms were assessed by study staff and led to collection of nasal fluid specimens for virological testing. The study objectives included investigating the response of respiratory symptoms to respiratory viral infection in children with and without asthma over a one year period. Due to sparse data daily respiratory symptom scores were aggregated into weekly average scores. More than 70% of the weekly average scores were zero, with the positive scores forming a skewed distribution. We propose a random effects probit/log-skew-normal mixture model to analyze the DAVIS data. The model parameters were estimated using a maximum marginal likelihood approach. A simulation study was conducted to assess the performance of the proposed mixture model if the underlying distribution of the positive response is different from log-skew normal.

Results: Viral infection status was highly significant in both probit and log-skew normal model components respectively. The probability of being symptom free was much lower for the week a child was viral positive relative to the week she/he was viral negative. The severity of the symptoms was also greater for the week a child was viral positive. The probability of being symptom free was smaller for asthmatics relative to non-asthmatics throughout the year, whereas there was no difference in the severity of the symptoms between the two groups.

Conclusions: A positive association was observed between viral infection status and both the probability of experiencing any respiratory symptoms, and their severity during the year. For DAVIS data the random effects probit -log skew normal model fits significantly better than the random effects probit -log normal model, endorsing our parametric choice for the model. The simulation study indicates that our proposed model seems to be robust to misspecification of the distribution of the positive skewed response.

Show MeSH
Related in: MedlinePlus