cosinor.poptests {cosinor2}

Comparison of Cosinor Parameters of Two Populations

Description

Runs the tests that compare MESORs, amplitudes and acrophases of two different populations.

Usage

cosinor.poptests(pop1, pop2)

Arguments

pop1

An object of the population.cosinor.lm class calculated on the first population.

pop2

An object of the population.cosinor.lm class calculated on the second population.

Details

Bingham et al. (1982) describe tests for comparing population MESORs, amplitudes and acrophases. These tests are esentially F-ratios with df_1 = m - 1 and df_2 = K - m, where m is the number of populations and K is the total number of subjects. The tests for MESOR, amplitude and acrophase differences respectively are calculated as follows:

F_M = \frac{\sum_{j = 1}^{m}k_j(\widehat{M}_j - \widehat{M})^2}{(m-1)\widehat{\sigma}_M^2}

F_\phi = \frac{\frac{\sum_{j = 1}^{m}k_j A_j^2 * sin^2(\widehat{\phi}_j - \tilde{\phi})}{m - 1}} {\widehat{\sigma}_\beta^2 sin^2\tilde{\phi} + 2\widehat{\sigma}_{\beta \gamma} cos\tilde{\phi}sin\tilde{\phi} + \widehat{\sigma}_\gamma^2 cos^2\tilde{\phi}}

F_A = \frac{\frac{\sum_{j = 1}^{m}(\widehat{A}_j - \widehat{A})^2}{m - 1}}{\widehat{\sigma}^2_\beta cos^2\widehat{\phi} - 2\widehat{\sigma}_{\beta \gamma}cos\widehat{\phi}sin\widehat{\phi} + \widehat{\sigma}^2_\gamma sin^2 \widehat{\phi}}

where \widehat{M}, \widehat{A} and \widehat{\phi} are weighted averages of parameters across populations calculated as:

\widehat{M} = \frac{\sum_{j = 1}^{m}k_j\widehat{M}_j}{K}

\widehat{A} = \frac{\sum_{j = 1}^{m}k_j\widehat{A}_j}{K}

\widehat{\phi} = \frac{\sum_{j = 1}^{m}k_j\widehat{\phi}_j}{K}

\tilde{\phi} is derived from the following expression:

tan 2\tilde{\phi} = \frac{\sum_{j = 1}^{m}k_j\widehat{A}^2_j sin 2\widehat{\phi}_j}{\sum_{j = 1}^{m}k_j\widehat{A}^2_j cos 2\widehat{\phi}_j}

where 2\tilde{\phi} lies between -\frac{\pi}{2} and \frac{\pi}{2} if the denomanator is positive or between \frac{\pi}{2} and \frac{3\pi}{2} if the denominator is negative, k_j is the number of subjects in the jth population, \widehat{M}_j, \widehat{A}_j and \widehat{\phi}_j are the cosinor parameters of the jth population and \widehat{\sigma}_\beta,\widehat{\sigma}_\gamma and \widehat{\sigma}_{\beta \gamma} are the estimates of pooled standard deviations (and covariance) calculated as following:

\widehat{\sigma}_\beta = \frac{\sum_{j = 1}^{m} (k_j - 1)\widehat{\sigma}_{\beta_j}}{K - m}

\widehat{\sigma}_\gamma = \frac{\sum_{j = 1}^{m} (k_j - 1)\widehat{\sigma}_{\gamma_j}}{K - m}

\widehat{\sigma}_{\beta \gamma} = \frac{\sum_{j = 1}^{m} (k_j - 1)\widehat{\sigma}_{{\beta_j} {\gamma_j}}}{K - m}

where \widehat{\sigma}_{\beta_j}, \widehat{\sigma}_{\gamma_j} and \widehat{\sigma}_{{\beta_j} {\gamma_j}} are the standard devations and covariance of \beta and \gamma parameters in the jth population.

Note

These tests should only be performed on independent samples. If the acrophases of two populations are significantly different, the results of the amplitude difference test are not reliable and should not be interpreted. While it's possible to perform tests which compare more than two populations, this function can only compare two populations.

References

Bingham, C., Arbogast, B., Guillaume Cornélissen, G., Lee, J.K. & Halberg, F. (1982). Inferential Statistical Methods for Estimating and Comparing Cosinor Parameters. Chronobiologia, 9(4), 397-439.

Examples

fit.extraverts<-population.cosinor.lm(data = PA_extraverts, time = PA_time,
period = 24)
fit.introverts<-population.cosinor.lm(data = PA_introverts, time = PA_time,
period = 24)
cosinor.poptests(pop1 = fit.extraverts, pop2 = fit.introverts)