I want to analyze the math score of high school students between 2009 and 2014 in the downtown of a city. The aim of the investigation is to identify the factors that affect the math score. In the methodology I am going to use the following models:

Linear regression and Logistic regression

Suppose that the math scores fall between 0.0 and 10.0 , for the logistic regression model if the math score is greater than or equal to 6.0, the score is assigned to class 1 (approved) otherwise the math score is assigned to class 0 (reprobate)

Finally I am going to compare those models.

Is correct to use logistic regression in this problem? Is there any loss of information and how can it affect the conclusions of the research?

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    $\begingroup$ Are the scores yes/no or continuous values? $\endgroup$ – gung - Reinstate Monica Sep 10 '15 at 0:47

The main difference between linear regression and logistic regression is the type of the response variable. Linear regression assumes the response variable is continuous (e.g. SAT scores, anywhere from 0 to 2400). Logistic regression assumes it is binary (zero or one; success or failure). Because I assume the test scores are continuous rather than just pass/fail, my answer to "Is correct to use logistic regression in this problem?" is "no."


The linear regression models an output variable $y$ as $$ y \sim N(\mu=\beta x, \sigma^2=\epsilon^2), $$ i.e. the output variable for input $x$ is normally distributed with a mean $\beta x$, with symmetric gaussian errors around this value. This goes beyond simply assuming a continuous output. In particular, it assumes that the output is unbounded. Since, in your case, scores are explicitly bound between $0.0$ and $10.0$, this assumption breaks down near the boundaries. On the other hand, logistic regression models an output variable $y$ as $$ y \sim B(1,p=g(\beta x)), $$ e.g. a binomial distribution with probability $\beta x$. So, logistic regression is actually just a specific type of binomial regression. (If we could rename regression techniques, it would be good to call OLS linear regression normal regression).

Neither of these completely captures your case, where you have either an ordinal output between 0 and 10 or a continuous output between 0 and 10. Are the scores all integer multiples, or half-integer multiples? You may want to consider a multi-level logistic regression, but this likely would be overly complex. An alternative is beta regression, modeling an output $$ y \sim B(\alpha=g(A x),\beta=g(B x)) $$ In this case, the output variable $y$ is continuous in the range $y \in [0,1]$. You can map your problem to this trivially by dividing the score by 10.0 so that it is a score out of 100%.


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