Joseph McDonnell made an interesting point when he slightly modified the
question.As I understand what he is asking, the question is "how far wrong
do we go when we treat the dichotomous dependent variable as if it were
continuous.
Cox and Wemuth (American Statistician, 1992) approached a similar problem
from the point of view of binary responses and standard least squares
regression.
Cox and Wermuth pointed out that over the range of p = .20 to .80, linear
and logistic regression are extremely similar. That would suggest that if p
is not extreme there is no great loss due to using linear regression. They
also pointed out, however, that is such a case the maximum reasonable r^2
is .36, and that occurs when the points all fall at .20 and .80.
At 06:36 PM 11/6/00 +0100, Joseph McDonnell wrote:
>Gentlemen,
>
>I agree with both of you. Several correspondents had already pointed out
that a
>logistic regression approach would be more appropriate in this situation.
>However, I was trying to steer the discussion in a slightly different
>direction,
>which I suspect may be the subject of Gerhard's interest (and of mine, but
>Gerhard can speak for himself). Perhaps I should rephrase the question to read
>'What are the consequences of performing an analysis on a discreet dependent
>variable as if it were continuous?' And I'm thinking of situations other than
>just linear/logistic regression. For example, in an artificial reproduction
>situation, the outcome may be clearly discreet (e.g. the number of IVF
>attempts)
>and numerically small enough for the discreteness to be (potentially)
important
>(most women only undergo relatively few attempts before dropping out/changing
>therapy). Not atypically, such studies are analysed using a Cox model, a
>continuous outcome model, using the number of attempts as 'survival time'. How
>valid are the conclusions from such an analysis? Under what circumstances does
>the analysis go up in smoke? etc. etc.
>
>Regards
>
>Joseph
>
>Jerry Dallal wrote:
>
>> Herman Rubin wrote:
>> >
>> > In article <[EMAIL PROTECTED]>,
>> > Joseph McDonnell <[EMAIL PROTECTED]> wrote:
>> > >I may be wrong, but I thought that Gerhard was asking something like
"If I
>> > >perform a linear regression but with a dichotomous dependent variable, do
>> > >I get 'garbage' results?"
>> >
>> > The results must be at least partly garbage. We can
>> > consider the dichotomous variable to be 0 or 1, and,
>> > using expectation, should interpret an answer between
>> > 0 and 1 as a probability.
>> >
>> > However, what meaning can be given to <0 or >1? The
>> > TRUE "linear regression" does not give the conditional
>> > expected value of the dependent random variable given
>> > the independent random variables, here the probability
>> > of 1, as it does in a linear model..
>> >
>> > >Joseph
>>
>> (1) Perhaps part of the problem is in the premise.
>> The subject header specifies "ANOVA".
>> With 2 groups, Student's t test applied to a binary outcome for
>> large samples is, for all practical purposes, the square root of
>> Pearson's chi-square statistic for homogeneity of proportions.
>> I have some simulation results for more than two groups which
>> suggests that ANOVA can be liberal in the 2xc case with nominal
>> levels of 0.05 being closer to 0.03.
>>
>> (2) Regression of a 0/1 variable on a continuous predictor is
>> Fisher's version of discriminant analysis. What Professor Rubin
>> says about the possiblity of misinterpretation is true, but that
>> doesn't invalidate the technique for use as Fisher intended.
>
>
>
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