ON REGRESSION AN INDICATOR VARIABLE In this technique, the dependent variable (Y) is an indicator, and takes a value of either 0 or 1. This is called a binary response variable
1 Response 0 Independent Variable(s)
Examples any two categories, any binomial or binary variable a) Success-failure, Gender (Male-female), mortality, presence-absence, pass-fail, etc. The results of a simple linear regression is a slope and intercept which will produce a predicted value which ranges from 0 to 1 over most of the range of X This b" can be interpreted as a probability of obtaining a 1 per unit of X, and the predicted value is the probability of obtaining a 1 at some particular value of X.
Problems with regression on indicator variables 1) Nonnormal errors : given that %3 = Y3 "! "" X3 , then When Y3 = 1, then When Y3 = 0, then 2) Nonconstant errors Let P(Y3 =1) = 13 and P(Y3 =0) = 1 13 then E(Y3 ) = 1(13 ) + 0(1 13 ) = 13 = "! "" X3
2 and 5]3 = E[Y3 E(Y3 )]2 = (1 13 )2 13 + (0 13 )2 (1 13 )
%3 = 1 "! "" X3 %3 = "! "" X3
= 13 (1 13 ) = E(Y3 )(1 E(Y3 )) finally, Var(%3 ) = Var(Y3 ), since %3 = Y3 13 and 13 is a constant
2 so 5%3 = 13 (1 13 )
= E(Y3 )(1 E(Y3 )) = ("! "" X3 )(1 "! "" X3 ) and the variance is a function of X3 3) Constraints on the response function If the function is fitted with a line, at some point the predicted value will be <0 or >1. As a probability, the true value must be between 0 and 1, so we must place some restraint on the predicted value.
So we would like to find a function which solves some of these problems, we might also expect a curve instead of a simple linear and we would like a curve that can from go 0 to 1 (asymptotically) Several sigmoid possibilities have been considered, especially a) Logistic (symmetric) b) cumulative normal distribution (Probit analysis) This version of the logistic has several advantages, E(Y) =
exp("! +"" X3 ) 1+exp("! +"" X3 )
particularly that it can be readily linearized by the transformation
1 1w = log/ 11
This is called a LOGIT transformation, and 1w is called a logit mean response. We can then fit
w 13 = b! + b" X3
and we should closely approximate the logistic. The logistic can also be fitted directly with nonlinear techniques.
A similar, but more difficult and less flexible, transformation exists for the cumulative normal distribution, and is called a PROBIT transformation
Weighting to improve variance : the logit only linearizes the logistic function, it does not cure the nonhomogeneous variance problem The logit,
1 1w = log/ 11
is estimated by
p pw3 = log/ 13p3
The variance of p3 is Var(pw3 ) =
1 n3 13 (113 )
which is estimated by, spw3 =
1 n3 p3 (1p3 )
we could therefore weight by w3 = n3 p3 (1 p3 ) in order to homogenize the variance.
Notes: 1) logits are readily extendible to multiple regression. 2) Logistic regression has many applications. One common application in the biological sciences is the calculation of the dose needed to cause mortality. However, small doses cause small mortalities and large doses cause large mortalities. We therefore calculate an LD&! , which is the lethal dose for 50% mortality".
for example, given the equation below ^w 13 = b! + b" X3 = -2.64 + 0.673*dose the LD50 is given by
50 ^w 1&! = log/ 150 = 0
0 = -2.64 + 0.673*dose&! dose&! =
2.64 0.673
= 3.923, or a dose of about 4