# What logistic regression is best to use?

My project is about whether early language scores (on a reading task) can predict later scores on a social attribution task. I have to conduct a logistic regression for my project but I am stuck on which one is appropriate. This is because I have conducted a binary logistic and multinomial logistic regression but I always get warnings and I do not know what to do. Any help will be much appreciated.

It's just that my supervisor specifically instructed me to use a logistic regression, sorry I forgot to add that there are three groups involved autism, language impaired and typically developing kids and I need to look at whether langauge scores of the autistic kids and language impaired kids are also a signifcant predictor of scoring low on a social attribution task in later life. Please can you help with this??

It sounds like either your professor gave you an incorrect recommendation, or you misunderstood, or you have not communicated what you want in a clear way. Something is wrong, in any case.

If you are trying to predict a numeric score (like IQ, SAT, etc) then you do not want logistic regression of any type. You probably want linear regression.

But perhaps that is not what you want to do. Perhaps you want to use the scores to predict the group that the person is in; since there are three groups, you do not want "ordinary" or binomial logistic regression, but you might want either ordinal or multinomial logistic, depending on whether the outcome is ordered or not.

One additional possibility is that your professor expects you to turn the numerical value of score into a categorical one; this is nearly always a bad idea, but it is often done, nonetheless. In this case, you would probably want ordinal logistic regression

If you are using SAS, you may find a presentation of mine informative; it is at www.nesug.org/Proceedings/nesug10/sa/sa03.pdf

• Thank you for very much for your help. much appreciated!!
– Miranda
Dec 29, 2010 at 15:28

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))


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.

• Thank you very much! It's just that my supervisor specifically instructed me to use a logistic regression, sorry I forgot to add that there are three groups involved autism, language impaired and typically developing kids and I need to look at whether langauge scores of the autistic kids and language impaired kids are also a signifcant predictor of scoring low on a social attribution task in later life. Please can you help with this??
– Miranda
Dec 26, 2010 at 22:29
• Thank you for very much for your help. much appreciated!!
– Miranda
Dec 29, 2010 at 15:30

Good explanation. Remember in your program output for the multilogistic model will only give estimates for two models even though there are three categories. The model that produces the largest estimate has the higheset odds from coming from that group. The last group can be determined by 1-estimate1-estimate2. From the top of my head, I think the following holds: exp(model1)/(1-exp(model2))=chance of model 1 and exp(model2)/(1-exp(model2))=chance of model 2.

• I think most statistical packages will output sth like $\log\big(\Pr(Y=2)/\Pr(Y=1)\big)=b_{10}+\sum_ib_{1i}x_i$ and $\log\big(\Pr(Y=3)/\Pr(Y=1)\big)=b_{20}+\sum_ib_{2i}x_i$; and the interpretation of a regression coefficient is straightforward: for a one-unit increase in a (continuous) $x$, the log of the ratio of either of these probabilities will be $b$.
– chl
Dec 28, 2010 at 8:16
• Thank you for very much for your help. much appreciated!!
– Miranda
Dec 29, 2010 at 15:29