RCIRCULAR procedure

Does circular regression of mean direction for an angular response (P.W. Goedhart).

Options

Parameter

Description

Procedure RCIRCULAR can be used to fit a circular regression model to an angular response. A circular regression model is similar in spirit to a generalized linear model; it employs the von Mises distribution and the arctangent link function. More formally, it is assumed that the angular response follows a von Mises distribution with mean direction μ and concentration parameter κ. The mean direction μ is related to the linear predictor η by means of the link function

μ = μ₀ + 2 arctan(η)

which maps the real line to the circle. The linear predictor η itself is a linear function of all the regressors in the usual way, except that it does not include a constant term. The circular regression model is fitted by means of an iterative algorithm which employs re-weighted least squares to estimate the linear parameters. A detailed account can be found in Fisher (1993) or Fisher & Lee (1992).

   Note that the model is not invariant to linear shifts of explanatory variates. This is because the linear predictor η does not contain a constant term. This can be a serious drawback of the circular regression model. An alternative model without the parameter μ₀ and including an intercept in the linear predictor is not invariant to rotations of the response, which is even worse. Also note that the estimates on page 161 of Fisher (1993) are for the centred distance explanatory variable.

   A call to RCIRCULAR must be preceded by a MODEL statement which defines the angular response variate. Only the first response variate is analysed and options other than WEIGHTS should not be set in the MODEL statement. The TERMS parameter of RCIRCULAR specifies the model to be fitted. Cases with a missing response variate or with a zero weight are excluded from the analysis. The FACTORIAL option operates in the usual way. Printed output is controlled by the PRINT option with the usual settings. Setting PRINT=summary displays the value of minus twice the maximized log likelihood, both for the fitted model and for the null model with only the constant μ₀. The difference between the two log likelihood values is also printed with a corresponding probability based on the chi-square distribution using likelihood ratio testing. This tests whether the fitted model is an improvement over the null model. PRINT=monitoring displays monitoring information of the iterative algorithm. The iterative process itself is controlled by the MAXCYCLE option which determines the maximum number of cycles, and by the TOLERANCE option. The iterative process is stopped when the relative difference in minus twice the log likelihood is smaller than the specified tolerance.

   Results of the circular regression can be saved by a number of options. The ESTIMATES, SE and VCOVARIANCE options save estimates of the linear parameters, their standard errors and variance-covariance matrix. This never includes the constant parameter. The estimate and standard error of the constant parameter μ₀ can be saved using options MU0 and SEMU0, and those for the concentration parameter κ of the von Mises distribution can be saved using options KAPPA and SEKAPPA. The _2LOGLIKELIHOOD option allows minus twice the maximized log likelihood to be saved, and the DF option saves the residual degrees of freedom. These may be useful for comparing a sequence of nested models fitted by RCIRCULAR using likelihood ratio testing. The RESIDUALS, FITTEDVALUES, LEVERAGES, ITERATIVEWEIGHTS, LINEARPREDICTOR and YADJUSTED options allow you to save the simple residuals, the fitted values (i.e. the fitted mean directions), the leverages of the iterative reweighted least squares algorithm, the linear predictor and an adjusted dependent variate. Finally the contribution of each unit to minus twice the maximized log likelihood can be saved by means of the I_2LOGLIKELIHOOD option.

Options: PRINT, FACTORIAL, RESIDUALS, FITTEDVALUES, LEVERAGES, ESTIMATES, SE, VCOVARIANCE, MU0, SEMU0, KAPPA, SEKAPPA, _2LOGLIKELIHOOD, DF, ITERATIVEWEIGHTS, LINEARPREDICTOR, YADJUSTED, I_2LOGLIKELIHOOD, MAXCYCLE, TOLERANCE.

Parameter: TERMS.

Method

The model is fitted using the algorithm of Fisher & Lee (1993) and Fisher (1993). The iterative fitting of the model is adapted by adding the linear predictor from the previous cycle to the adjusted y variate. For a weighted circular regression the estimated circular standard error of μ₀ is calculated using the sum of the weights instead of the degrees of freedom, see equation (6.64) in Fisher (1993). Note that the estimated standard errors for the linear parameters are conditional on the estimates of μ₀ and κ, and vice versa.

Action with RESTRICT

Only the angular response variate can be restricted. The analysis is restricted accordingly.

References

Fisher, N.I. & Lee, A.J. (1992). Regression models for an angular response. Biometrics, 48, 665-677.

Fisher, N.I. (1993). Statistical Analysis of Circular Data. Cambridge University Press, Cambridge.