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RE: [Spam:0007 SpamScore] Re: Graphs
Ah, Dr. Johnson, sorry you have not been feeling so well. Please get better
soon, so that you are able to enjoy the break!
>===== Original Message From PS707-L_AT_listproc.cc.ku.edu =====
>Dear Linsey:
>
>I've had several questions like yourss during the week and I'm beginning
>to understand what you-all don't understand. I'm broadcasting the answer
>to the list, to find out if I'm saying the same thing to all of you and
>also with the hope that I'll remember the answer in the future.
>
>THe pictures you have don't do any good because they just help you see
>that you have data ill suited to a Poisson. We know that already, but we
>want to go through the motions as if it were suitable.
>
>You need to get predicted values for some range of X and plot through
>the points. Its exactly the same exercise as logistic regression.
>
>mymod <- glm(Y~X,family=poisson)
>
>Suppose X ranges from 1 to 10
>
>pred1 <- predict(mymod, newdata=data.frame(X=1:10), type="response")
>
>plot(X,Y, type="n")
>points(X,Y)
>lines(1:10, pred1)
>
>Why do this?
>
>Describe OLS. The thing we use as a predicted value is the expected
>value of Y given X:
>
>E(Y|X) = a + b X
>
>and to account for variety in Y, we say
>
>Y ~ Normal( a+b X, sigma^2)
>
>Now, if you have a count variable, the default formula used in glm with
>the Poisson family has an expected ("mean") value
>
>E(Y|X) = exp (a + b x)
>
>Note if you log both sides you have
>
>log( E(Y|X)) = a + b x
>
>For some reason, the pioneers of these models were emphatic about
>thinking of the transformation in that way, as a fudge on the mean
>prediction. That is, they wanted to talk about the transformation from
>the "mean" back to the linear predictor. They call that a link function.
>
>This is the curved "line" you see if you plot predictions. In glm, you
>can get that with glm(Y~X,family=poisson(link=log)) which is the
>default, so if you just do glm(Y~X,family=poisson) then you get the same
>thing.
>
>But the distribution of Y is different, it is Poisson
>
>Y ~ Poisson( exp(a+ b X) )
>
>The use of the exp there translates the a+bX into the mean of Y. The
>Poisson has the property that its expected value equals its variance
>equals its "lambda" parameter, the one I called "input" on my handout.
>The exp() is used for a variety of reasons. One really big reason is
>that the value of (a + bX) has to be 0 or greater. Otherwise, Poisson
>distribution is not defined. (Poisson does not exist for negative
>numbers). You can find other transformations besides exp() that keep
>the value of (a+bX) positive, they are OK. You can even chance it and
>ignore the problem and run a Poisson with
>
>E(Y|X) = a + b X.
>
>If you want that, do glm(Y~X,family=poisson(link=identity))
>
>So if you want to compare the Normal model against Poisson, you should
>transform the equation for the Normal model so that the expected value
>is the same. If you did OLS with this assumption
>
>E(Y|X) = exp(a + b X)
>
>Then the predicted values of the 2 models would match if their a and b
>were the same. You'd estimate that in lm by
>
>mymod<-lm ( log(Y)~ X )
>
>Get predicted values like always, except: THe predicted values from that
>would be in logged values, and you'd need to translate back into the Y
>scale with
>
>predY <- exp(predicted_lm_values)
>
>
>
>Then, if you get the predicted values out of the Poisson model with the
>predict function (don't forget to add the predict option type="response")
>
>Then you make a plot with the raw data points, and overlay the 2 kinds
>of lines.
>
>In the example that Nathan had, the 2 lines were quite close.
>
>Of course, the distribution of points that is implied by a model depends
>on the distribution. SO with your eye, you can decide for yourself if
>the Normal or Poisson seems more "right". Normal would have points
>evenly divided on either side of the line, and OLS requires that
>homoskedasticity as well.
>
>On the other hand, if the predicted value is small, then the
>distribution of Y in a Poisson is quite not normal.
>
>
>In the days before VGA graphics, we had to do all artwork with letters,
>so this takes me back:
>
>XX
>XX
>XX XX
>XX XX
>XX XX XX
>XX XX XX XX
>XX XX XX XX XX
>
>
>As the predicted value goes up, the distribution of Y predicted by
>Poisson looks more and more normal.
>
>
> XX
> XXXXXX
> XXXXXXXXXXX
> XXXXXXXXXXXXXXXXXX
>
>If your expected value is a big number, say 100, then whether it is
>Poisson or Normal is not very important. However, for a small predicted
>value, then there is a huge difference. Doesn't my beautiful ascii art
>show it???
>
>
>
>I'm not feeling so well for the past few days, otherwise I'd offer to
>come to work and help you. But if you questions about how to make R do
>your work, you can feel free to ask me in ps707-l.
>
>
>
>linseym wrote:
>> Prof. Johnson, I just tried emailing you this so disregard this message if
you
>> already got it, the computer said it was having problems with the webmail
>> connection. So here goes, I ran the Poisson graphs but have no clue how to
>> interpret them (remember my dep.var. is the aweful 4 ordinal categories)
If
>> you can please help.
>> Thanks, Linsey
>>
>> I have attached the graphs for you to look at,
>> they look really cool.
>
>
>--
>Paul E. Johnson email: pauljohn_AT_ku.edu
>Dept. of Political Science http://lark.cc.ku.edu/~pauljohn
>1541 Lilac Lane, Rm 504
>University of Kansas Office: (785) 864-9086
>Lawrence, Kansas 66044-3177 FAX: (785) 864-5700