Question

In a certain country 30% are seniors, and 40% of the seniors are vitamin D deficient, and among the non-seniors, 20% are vitamin D deficient. If a person is chosen randomly, let X be the age (1=senior, 2=not senior), and let Y be the vitamin D status (1=deficient, 2=not deficient).

(a) Let πij = P(X=i;Y=j). Obtain the table for the joint probabilities {πij}

(b) Find the odds for vitamin D deficiency among seniors and among non-seniors. Also obtain the odds ratio for vitamin deficiency for seniors to non-seniors. Interpret this odds ratio.

(c) Are age and vitamin D status independent? Explain your answer.

Answer 1

Given: Pr(Senior) = 0.3 and Pr(Non-senior) = 0.7

(a)  Let πij = P(X=i;Y=j).The joint probability table can be constructed as follows:

Here, The marginal probabilites of seniors and non seniors are 0.3 and 0.7 respectively, as given in the data.Next, it is given that  40% of the seniors are vitamin D deficient, i,e 40% of 30% are vitamin D deficient.So, seniors being vitamin D deficient has the probability 40% of 30% = 0.4 * 0.3 = 0.12.

Also,  among the non-seniors, 20% are vitamin D deficient, i.e. 20% of the 70%. So non seniors being vitamin D deficient has the probability 20% of 70% = 0.2 * 0.7 = 0.14.

The remaining probabilites for the 'Not deficient' class can be obtained by subtacting those for the 'Deficient class' from the marginal probabilities.

Seniors who are not vitamin D deficient = 0.3 - 0.12 = 0.18 Non seniors who are not vitamin D deficient = 0.7 - 0.14 = 0.56

(b) Odds  for vitamin D deficiency among seniors is nothing but the ratio of probability that the seniors are deficient to the probability that the seniors are not.

Odds for vitamin D deficiency among seniors:

Odds for vitamin D deficiency among seniors = Odds (Seniors) = Pr(Seniors are vitamin D deficient) / Pr(Seniors are not vitamin D deficient)

=

Odds for vitamin D deficiency among non seniors = Odds ( Non seniors)

= Pr (Non seniors are vitamin D deficient) /Pr(Non seniors are not vitamin D deficient) =   

Odds ratio is nothing but ratio of Odds (Seniors) to the Odds ( Non seniors) =

The figure 2.667 implies that, for seniors, the odds of being vitamin D deficient are 2.667 times larger than that for non seniors.

(c) We know that Odds ratio of 1 indicates that the 2 factors, here, Age and Vitamin D status are independent.But here, the odds ratio is significantly different from 1.

Also, the factors in the contingency table are independent if the joint probability is equal to the product of their marginal probabilities.

Here, P(Seniors and Deficient) = 0.12 P(Seniors) *P(Deficient) = (0.3)(0.26) = 0.078  

Since,  P(Seniors and Deficient)   P(Seniors) *P(Deficient), it is evident that Age and Vitamin Status are dependent.

Similarly,  P(Non seniors and Deficient)   P(Non seniors) *P(Deficient) P(Seniors and Not deficient)   P(Seniors) *P(Not deficient)   P(Non seniors and Not deficient)   P(Non seniors) *P(Not deficient)

Hence Age and Vitamin D status are not dependent.