Logistic regression is to be used when the outcome of interest is a binary variable (e.g., success/failure), whereas multinomial logistic is reserved for the case of a multi-category response variable (e.g., blue/red/green). In both cases, the response variable to be predicted is a categorical variable. The predictors might be categorical or continuous variables.
From what you described, you are interested in predicting scores on a social attribution task based on observed scores on a reading scores. If both sets of scores are numerical, then it is simply a linear regression model. In R, it is something like
# fake data
x <- rnorm(100)
y <- 1.2*x + rnorm(100)
# model fit
summary(lm(y ~ x))
The output reads:
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-2.6635 -0.6227 -0.1589 0.6360 2.2494
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.03769 0.10042 -0.375 0.708
x 1.31590 0.10796 12.188 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.003 on 98 degrees of freedom
Multiple R-squared: 0.6025, Adjusted R-squared: 0.5985
F-statistic: 148.6 on 1 and 98 DF, p-value: < 2.2e-16
Here an increase of one unit in $x$ is associated to an increase of 1.32 units in $y$. This is basically what the regression coefficient tells you when you assume a model like $\mathbb{E}(y|x)=\beta_0+\beta_1x+\varepsilon$, where $\varepsilon$ are independent and identically distributed as a centered gaussian with variance $\sigma^2$ (unknown).
Now, you may still be interested in a binary outcome. For instance, I can define "Success" as $y>0$. In this case, a logistic regression would look like
yb <- ifelse(y>0, 1, 0)
summary(glm(yb ~ x, family=binomial))
where the regression coefficient gives you the log of the odds of passing the exam (compared to failing to reach a score of 0):
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.5047 0.3194 -1.580 0.114
x 3.3959 0.6871 4.943 7.7e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Edit
Some pointers with multinomial logistic regression have already been given. What I wonder is the outcome you want to model: If you're interested in whether the autistic group has lower scores on the attribution task (as would be expected from the literature), then in a logistic model both scores will be continuous predictors, and reading scores will be considered as a covariate; in other words, you model the odds of being in one of the diagnostic class as a function of attrition task, after adjusting for baseline differences in language proficiency. But it seems it would make sense only if the diagnostic categories are not a priori defined and it does not directly answer the question you are asking (whether language is a predictor for attrition that may act differentially according to the diagnostic group); otherwise, I would rather model the attrition scores as a function of a grouping variable (diagnostic category) + reading scores, which is basically an ANCOVA model.
Both models are available in R: mlogit() in the package mlogit for multinomial logistic regression, and lm() for ANCOVA.
