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CONDITIONAL AND MARGINAL MEANS
I. Question : How are conditional means E(Y|X) and marginal means E(Y) related? Simple example : Population consisting of n 1 men, n 2 women. (So size of entire population is n 1 + n 2 .) Y = height X = sex Categorical, two values: Male, Female Conditional means: E(Y|male) = (Sum of all men’s heights)/n 1 E(Y|female) = (Sum of all women’s heights)/n 2 Rearrange: Sum of all men’s heights = n 1 E(Y|male) Sum of all women's heights = n 2 E(Y| female) Marginal mean E(Y) = (Sum of all heights)/(n 1 + n 2 ) = ! ( Sum of men ' s heights ) + ( Sum of women ' s heights ) n 1 + n 2 = ! n 1 E ( Y | male ) + n 2 E ( Y | female ) n 1 + n 2 = n 1 n 1 + n 2 E(Y|^ male)^ +^ n 2 n 1 + n 2 E(Y|^ female) = (proportion of males)( E(Y| male) + (proportion of females)( E(Y| female) = (probability of male)( E(Y| male) + (probability of female)( E(Y| female) Thus : The marginal mean is the weighted average of the conditional means, with weights equal to the probability of being in the subgroup determined by the corresponding value of the conditioning variable.
X categorical with more than 2 values : If the population consists of m subpopulations pop 1 , pop 2 , …, popm (equivalently, if we are conditioning on a categorical variable with m values - - e.g., the age of a fish), then E(Y) = Pr( popk ) E ( Y | popk ) k = 1 m
e.g., fish: popk = fish of age k, so E(length) =
Pr( Age = k ) E ( Length | Age = k )
k = 1 6
Rephrase: If the categorical variable X defines the subpopulations, E(Y) = Pr(^ x ) E ( Y^ |^ X^ =^ x ) all values x of X
In words: X Continuous :
E(Y) = "# fX^ ( x ) E^ (^ Y^ |^ x )^ dx
fX(x) = probability density function (pdf) of X. Note:
- There are analogous results for conditioning on more than one variable.
- The analogous result for sample means is
y =
This reasoning can be generalized to give: The expected value of the conditional means is the weighted average of the conditional means, which by Part 1 is the marginal mean: E(E(Y|X)) = E(Y)
III. CONDITIONAL AND MARGINAL
VARIANCE
Marginal Variance : Definition of (population) (marginal) variance of a random variable Y: Var(Y) = E([Y - E(Y)]^2 ) In words and pictures: Useful formula for Var(Y): Var(Y) = E([Y - E(Y)]^2 ) = E(Y^2 - 2 YE(Y) + [E(Y)]^2 ) = = In words:
Conditional Variance : Similarly, we define the conditional variance Var(Y|X) = Variance of Y|X = E([Y - E(Y|X)]^2 | X) Note: Both expected values here are conditional expected values. In words: Additional formula for conditional variance: Var(Y|X) = E(Y^2 |X) - [E(Y|X)]^2 (Derivation left as exercise for student.) Conditional Variance as a Random Variable : Var(Y|X) is a random variable. Example: Y = height, X = sex for persons in a certain population Var(height | sex) is the variable which assigns to each person in the population the variance of height for that person's sex.
2 ) () implies Var(Y)! E(Var(Y|X)) (Why?) Moreover, Var(Y) = E(Var(Y|X)) if and only if Var(E(Y|X)) = 0. This says: 3 ) Since Var(Y|X)! 0 , E(Var(Y|X)) must also be! 0. (Why?). Thus () implies Var(Y)! Var(E(Y|X)). Moreover, Var(Y) = Var(E(Y|X)) if and only if E(Var(Y|X)) = 0. Since Var(Y|X)! 0 , E(Var(Y|X)) = 0 says that for each value of X, Var(Y|X) = ____. This implies that for each value of X, Y|X is ______________. Thus the relationship between Y and X is __________________. 4 ) Another perspective on () (cf. Textbook, pp. 36 - 37 ) i) E(Var(Y|X) is a weighted average of Var(Y|X) ii) Var(E(Y|X) = E([E(Y|X) - E(E(Y|X))]^2 ) = E([E(Y|X) - E(Y)]^2 ), which is a weighted average of [E(Y|X)- (E(Y)]^2 Thus, () says: Var(Y) is a weighted mean of Var(Y|X) plus a weighted mean of [E(Y|X) - E(Y)]^2 - - and is a weighted mean of Var(Y|X) if and only if all conditional expected values E(Y|X) are equal to the marginal expected value E(Y).)
WHAT CONTRIBUTES MOST TO VAR(Y):
VAR(E(Y|X)) OR E(VAR(Y|X))?
A. 012345678 910 10 5 0 x 0 2 4 6 8 10 15 10 5 0 y B. 012 345678910 5 0
- 5 x - 5 - 3 - 1 1 3 5 7 20 10 0 y C. 0 5 10 10 5 0 x - 2 0 2 4 6 8 10 12 10 5 0 y
