high

P Dixon
JME 59 (2008) 447-456

Use logistic repression for dichotomous data with properties such as thsoe of accuracy (Allison, 1999; Everitt, 2001).

logit (P(C)) = a + b1X1 + b2X2 + ...

Logistic repression is described as appropriate for the analysis of dichotomous data when there are tow possbile responses and several continuous or categorical predictors... McCullagh(1980): logistic regression models are appropriate when the categorical responses can be construed as contiguous intervals on a continuous scale.

Luce (1963) choice theory - correct and incorrect responses are asociated with response strengths... the probability of selecting the correct response is given by the ratio of the correct response strength to the sum of the response strengths:

P(C) = Sc/ Sc + Se

... with some arrangement, logit(P(C)) = ln [Sc/Se] = ∑Ψi
where Ψi = ln(Ψc,i/ Ψe,i),
the logit is a linear function of the processing components that determine the relative response strength. Experimental factors that affect components selectively will thus have additive effects in a logistic regression equation.

key phrases: accuracy scale is bounded;


-- the pattern of means may not provide an informative reflection of the underlying processes if the levels of accuracy approach those bounds.

as the level of accuracy increases, effects that are in principle additive may appear to exhibit an underadditive interaction, while data that derive from an overadditive interaction may appear to be additive.

p451- reasons against ad-hoc arcsine transformation
* applying an arcsine transformation to the means reduces this value to about 23%, but misleading interpretations would still be possible;
* the transformed means do not necessarily have a simple interpretation in terms of the underlying mechanisms;
* because the choice of transformation is ad hoc, it may be difficult to defend that choice if it has a large effect on the pattern of means;

Although logistic regression provides a reasonable approach to the analysis of accuracy data, it cannot be readily applied to data from repeated-measures designs. - this is because in standard logistic regression it is assumed that all of the observations in the design are independent.

Solutions:
1. Conditional Logistic Regression
2. GLM mixed effects models

Conditional Logit Regression

The probibility of person i responding correctly to item j is assumed to be an inverse logistic function of a person parameter and na item difficulty parameter:

P(C) = logit(-1 superscript)(θi - βj).

To apply this development in the present context, one can construe the item parameter as a linear function of the experimental factors:

P(C) = logit(-1 superscript) (θi - (μ + αj + βk + ...))

Rasch model is identical to the logistic repression model but with the addition of a random subject term.

A number of stats packages can perform the relevant conditional logistic regression, including the clogit program in the survival package in R.

Rationale for using conditional logit regression -- random effect of subjects is limited to an overall variation in performance and does not interact with the effects of interest. After conditionalising on the random contribution of subjects, each of the observation s can be assumed to be independent, and the interpretation of the logistic regression parameters can proceed as before.

2. Generalised linear mixed-effects models
although condiitonal logistic model incorporates the assumption that subjects are randomly sample, the approach fails to address situations in which the magnitude of an effect varies over subjects. This can be addressed by using mixed-effects modelling. LME can be used in GLM, allowing one to fit logistic regression models with random effects.

lmer (Obs ~ A + (1|S), family = binomial) --> random intercept given each value of S
lmer (Obs ~ A + (1+A|S), family = binomial) --> random slope of factor A across subjects