Supplementary method and results

Alternative statistical analysis for General Linear Models where homogeneity of slopes assumption was violated- The Wilcox modification of Johnson-Neyman procedure.

Analysis of Covariance (ANCOVA) is the statistical procedure which compares mean values of response variable between groups when the response (dependent) variable covaries with other continuous variables (covariates).

It is a variant of General Linear Models and the basic design for one-factor fixed effect ANCOVAis defined by the equation:

Yij=μ + αi +β(Xij-) + ε

Where:

Yij=the values for the dependent response variable in each factor’s category;

μ = mean intercept;

αi=the response to the factor;

Xij= the individuals values of covariate for each category of the factor;

= the mean value of the covariate for all individuals;

β = the overall pooled regression coefficient (slope) within groups;

ε = the normally distributed error variance (Huitema 1980; Engqvist 2004).

Homogeneity of regression slopes in ANCOVA is one of the assumptions of this analysis. Violation of this assumption expressed as a significant factor-by-covariate interaction effect on the dependent variable, indicates that the relationship between the covariate and the response variable Y differs between groups.

When the factor-by-covariate is modelled, the significance test will not test for the average response to the factor, but for the differences in the Y intercept. The effect of the factor is not fixed but conditional, depending on the values of the covariate.

The original method to determine the covariate’s ranges for which a dependent response variable truly differs between groups, is the Johnson-Neyman technique (Johnson and Neyman 1936).

In our analysis, we used a modification of this method, the Wilcox procedure (Wilcox 1987). This is a post-hoc procedure testing over what ranges of the covariate the group means are different. For applying this method we used a program (WILCOX.exe) developed by A. Constable (Quinn and Keough 2002).

The Johnson-Neyman (J-N) procedure is analogous of testing simple main effects in MANOVA with a significant interaction effect (Huitema 1980). This is performed by determining the lower and upper limits of the covariate (X lower and X upper) between which there is 95% certainty that two investigated treatments are not significantly different. Wilcox (1987) developed formulae similar to the J-N procedure, which differ only in multiple comparison procedure and the type of used distribution. The advantage of the method is that accounts for unequal variances, allows for simultaneous determination of upper and lower limits of thenon-significance regions for all the pairs of treatments and subsequent generalizations. The mathematical formulae of the Wilcox procedure are extensively presented in Quinn and Keough 2002.

References:

  1. Huitema, BE. The Analysis of Covariance and Alternatives. New York: J. Wiley 1980.
  2. Engqvist, L. The mistreatment of covariate interaction terms in linear model analyses of behavioural and evolutionary ecology studies. Animal Behaviour 2005, 70:967-971.
  3. Johnson PO and Neyman J. Tests of certain linear hypotheses and their application to some educational problems. Statistical Research Memoirs 1936, 1:57-93.
  4. Wilcox RR. Pairwise comparisons of J independent regression lines over a finite interval, simultaneous pairwise comparison of their parameters, and the Johnson-Neyman procedure. British Journal of Mathematical and Statistical Psychology 1987 40:80-93.
  5. Quinn GP and Keough MJ. Experimental Design and Data Analysis for Biologists. Cambridge: Cambridge University Press 2002.

Supplementary results:

Supplementary table 2: Diagnosis and sex differences in entropy and uniformity
Dependent variable / Effect / F / Sig. / Partial Eta Squared (effect size)
Entropy unfiltered / Diagnosis / 6.94 / 0.011 / 0.124
Sex / 0.005 / 0.942 / 0.000
Diagnosis * Sex / 2.24 / 0.141 / 0.044
Uniformity unfiltered / Diagnosis / 7.485 / 0.009 / 0.133
Sex / 4.455 / 0.040 / 0.083
Diagnosis * Sex / 4.583 / 0.037 / 0.086
Entropy fine / Diagnosis / 0.842 / 0.363 / 0.017
Sex / 0.342 / 0.561 / 0.007
Diagnosis * Sex / 2.001 / 0.164 / 0.039
Uniformity fine / Diagnosis / 3.114 / 0.084 / 0.06
Sex / 1.091 / 0.301 / 0.022
Diagnosis * Sex / 0.08 / 0.778 / 0.002
Entropy mediumΔ / Diagnosis / - / - / -
Sex / - / - / -
Diagnosis * Sex / - / - / -
Diagnosis*Age / 4.127 / 0.022 / 0.147
Uniformity medium / Diagnosis / 3.139 / 0.083 / 0.06
Sex / 6.167 / 0.016 / 0.112
Diagnosis * Sex / 0.002 / 0.961 / 0.000
Entropy coarseΔ / Diagnosis / - / - / -
Sex / - / - / -
Diagnosis * Sex / - / - / -
Diagnosis*Age / 6.483 / 0.014 / 0.117
Uniformity coarse / Diagnosis / 3.705 / 0.06 / 0.07
Sex / 4.387 / 0.041 / 0.082
Diagnosis * Sex / 0.052 / 0.821 / 0.001

ΔFor these TPs the heterogeneity of slopes indicates that a usual ANCOVA is inappropriate and the diagnosis effect is conditional, depending on the age values. The ranges of age for which the diagnosis effect is truly significant were calculated with the Wilcox procedure.