Question

Please show me the R outputs with it

4. (15 pts) Hip dysplasia is a hip socket abnormality that affects many large breed dogs. A review of medical records of dogs seen at 27 veterinary medical teaching hospitals found that hip dysplasia was more common in Golden retrievers than in Border Collies; the data are shown in the following table. Border collie Golden retriever Hip dysplasia 3,995 42,946 46,941 221 Yes 5,007 5,228 No Total (a) Calculate the relative risk of hip dysplasia for Golden retriever compared to border collie (b) Calculate the sample value of odds ratio. (c) Construct a 95% confidence interval for the population value of the odds ratio (d) Interpret the confidence interval from the part (b) in the context of this setting.

Answer 1

(a)

From table:

P(Hip dysplasia in golden retriever) = 3995/ 46941

P(Hip dysplasia in border collie) = 221/ 5228

The relative risk is:

RR =P(Hip dysplasia in golden retriever) / P(Hip dysplasia in border collie) = ( 3995/ 46941) / ( 221/ 5228) = 2.01

Since RR is greater than 1 so it shows that risk of hip dysplasia is higher in golden retriever group.

(b)

Odds for golden retriever group:

$O_{1}=3995: 42946$

Odds for border collie group:

$O_{2}=221:5007$

Odds ratio:

$OR=\frac{O_{1}}{O_{2}}=\frac{3999/42946}{221/5007}=2.11$

(c)

The 95% confidence interval for odds ratio

$ln(OR)\pm 1.96\sqrt{\frac{1}{3995}+\frac{1}{221}+\frac{1}{42946}+\frac{1}{5007}}=ln(2.11)\pm 1.96\sqrt{\frac{1}{3995}+\frac{1}{221}+\frac{1}{42946}+\frac{1}{5007}}$

$=0.7467\pm 0.1386=(0.6081,0.8853)$

So required 95% confidence interval for odds ratio is

$\left (e^{0.6081},e^{0.8853} \right )=(1.84,2.42)$

(d)

Since confidence interval does not contain 1 so we cannot conclude that risk of hip dysplasia is same in both group.

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Following is the R script:

#Given information: a = 3995 b = 42946 C = 221 d = 5007 #part a: Risk in group golden retriever RR1 = a /(a+b) # Risk in group border collie RR2 = C/ (c+d) # Relative risk is RR = RR1 / RR2 RR #part b: odds for golden retriever group: 01 = a/b #odds for border collie group: 02 = c/d # odds ratio OR = round (01/02,2) OR # part c #95% confidence interval for odds ratio # lower limit 11 =round (exp( round(log (OR) - 1.96 sqrt((1/3995)+(1/221)+(1/42946)+(1/5007)),4)),2) # upper limit ul =round (exp(round(log (OR) + 1.96 sqrt((1/3995)+(1/221)+(1/42946)+(1/5007)),4)),2) # The required confidence interval is 11 ul

Following is the output:

> #Given information: > a = 3995 > b = 42946 > c = 221 > d = 5007 V V V > #part a: Risk in group golden retriever > RR1 = a /(a+b) > # Risk in group border collie RR2 = c/ (c+d) > # Relative risk is V V V V > RR = RR1 / RR2 > RR [1] 2.013297 > #part b: odds for golden retriever group: > 01 = a/b V V V V V V V V V V V #odds for border collie group: > 02 = c/d > # odds ratio > OR = round(01 /02,2) > OR [1] 2.11 > # part 0 #95% confidence interval for odds ratio V V > # lower limit V V 11 =round(exp( round(log(OR) - 1.96*sqrt((1/3995)+(1/221)+(1/42946)+(1/5007)),4)),2) V V # upper limit V > ul =round(exp(round(log(OR) + 1.96 sqrt((1/3995)+(1/221)+(1/42946)+(1/5007)),4)),2) > # The required confidence interval is [1] 1. 84 > ul [1] 2.42