27. Eye cataracts are responsible for over 40 percent of blindness around the wo

ID: 3364835 • Letter: 2

Question

27. Eye cataracts are responsible for over 40 percent of blindness around the world. Can drinking tea regularly slow the growth of cataracts? We can't experiment on people, so we use rats as subjects. Researchers injected 14 young rats with a substance that causes cataracts. Half the rats also received tea extract; the other half got a placebo. The response variable was the growth of cataracts over the next six weeks. The researchers found that the tea extract did slow cataract growth in the rats. (a) Outline the design of this experiment. (b) Use Table 7.1 (p. 246), starting at line 108, to assign rats to treatments.

Explanation / Answer

Part (a)

Design of Experiment

This will be a ‘Completely Randomised Design’ with 2 treatments and equal number of observations (7 each) per treatment.

Treatment 1: cataract-causing-substance injection + tea extract

Treatment 2: cataract-causing-substance injection + placebo

Response variable:

xij = growth of cataract over 6-week period of the jth rat assigned to treatment i, i = 1, 2 and j = 1, 2, …., 7.

Subject (rats) assignment to treatments: random using standard random numbers.

Model: xij = µ + i + ij, where µ = common effect, i = effect of ith treatment and ij = error which is assumed to be N(0, 2)

Analysis:

Let

xi. = sum of xij over j = 1 to 7

Grand Total, G = x.. = x1. + x2.

Correction Factor, C = G2/14

Total Sum of Squares SST = (sum of xij2 over j = 1 to 7 and i = 1, 2) - C

Row (Treatment) Sum of Squares SSR = {(x1.2 + x2.2)/7} - C

Error (Residual) Sum of Squares SSE = SST – SSR

ANOVA TABLE

Source of Variation

Degrees of Freedom (DF)

Sum of Squares (SS)

Mean Sum of Squares (MS)

Fcal

Fcrit

Treatment

2 – 1 = 1

SSR

MSR/MSE

F1,12,

Error

13 – 1 = 12

SSE

SSE/12

Total

14 – 1 = 13

SST

SST/13

DF: Treatment – Number of treatments – 1, Total – Total number of observations – 1.

Error – DF(Total) - DF(Treatment).

MS = SS/DF

F1,12, = upper % point of F-distribution with degrees of freedom 1 and 12; being the level of significance.

Reject the hypothesis i’s are equal if Fcal > Fcrit.

Part (b)

Assignment of rats to treatment by employing random numbers.

Process

Serially number the rats from 1 to 14.

Pick serially 2-digit random numbers from the directed table and line. Find mod 7 of the picked random number to get the serial number of the rat to be assigned to Treatment 1. Reject 0 and 99. Reject repeat numbers after mod conversion.

Actual assignment

Stage

Random number picked (n)

Mod 7 of n

Serial number of rat to Treatment 1

Thus, rats numbered 2, 3, 4, 6, 7, 10, and 12 are assigned to Treatment 1 and left-over rats are assigned to Treatment 2.

DONE