Statistical analysis of biological data (comparison of proportions)

Zagazig universityFaculty of Veterinary Medicine

Animal Wealth Development Department

Session#2:Statistical Analysis of Biological Data

(Comparison of Proportions)

M.Afifi

M.Sc., Biostatistics (Joint Supervision with ISSR, Cairo University) Ph.D., Candidate (AVC, UPEI, Canada)

E-mail: M.Afifi@zu.edu.eg, Afifi-stat6@hotmail.com Tel: +201060658185

Comparing of 2-Independent proportions

Testing associations

χ2

Assumptions Categorical variables, and data contained in the cells of the table are frequencies.

Animal independent in that no animal/ individual may be represented more than once

in the table.

No more than 20% of the cells of the table should have an expected frequency < 5.

If necessary:

• we can reduce the contingency table in size by combining appropriate rows

and/or columns.

• Alternatively, Fisher’s exact test on frequencies contained in a

Comparing two proportions

2 × 2 table, fourfold table, two-way frequency or contingency table

If there is no association between the outcome and the group, then we would

expect the proportions of successes to be the same in the two groups. Thus, we

can compare the two proportions by investigating the association between the

two factors that define the contingency table (outcome, group)

9.4.4 Example

The non-obese diabetic (NOD) mouse develops an autoimmune diabetes that can be used as a

model for human juvenile insulin-dependent diabetes. In the colony of Hawkins et al. (1993),

the incidences for male and female NOD mice were 24% and 73%, respectively. Hawkins et

al. investigated the causes of this sex difference by considering the effect of early castration

on the incidence of diabetes in male NOD mice. (The following is based on their findings.)

Fifty mice were randomly selected from 100 male mice and were castrated 1 day after birth;

they were compared with the remaining 50 sham-operated mice. The mice were maintained

for 140 days, and blood samples were collected biweekly starting at 42 days old. Diabetes was

determined by three consecutive blood glucose levels greater than 200 mg/dl. It was shown

that neonatal castration more than doubled the incidence of diabetes (52%) when compared

with controls (24%) at day 112. But is this difference significant?

Null hypothesis: that the two population proportions are equal orno association

between the two factors of interest. Specify the alternative hypothesis, generally

that the two proportions are not equal or that there is an association between the

two factors.

Test statistic: Chi-squared test, With Yates’ correction

P-value

Diabetes.sav

2 variables:

• Castration : Castrated, Non-castrated

• Diabetes: Diabetic, Non-diabetic

Analyze >>>> Descriptive Statistics >>>> Crosstabs

Yates’ correction, a continuity correction included to remove bias. This bias arises because we are assuming the test

statistic approximates the continuous Chisquared distribution although it has a

discrete distribution.

Computed only for the 2 × 2 table.

The 95% confidence interval for the true difference in the proportions of mice

with diabetes in the two groups is

χ2 value 7.17 with 1 degree of freedom, and (P = 0.007). >>>> reject null

hypothesis that the true proportions of mice with diabetes are equal in the control

and castrated groups.>>>>> There is evidence to indicate that neonatal

castration is linked with the incidence of diabetes in NOD mice.

Difference in incidence of diabetes in male and female mice may be associated

with the concentration of testosterone in the blood circulation.

> 2 proportions

Following staff training in insemination techniques in cattle, an artificial

insemination centre compared three training methods. The cows were

randomly assigned to a particular training method, each cow was inseminated

once and the proportion of cows that became pregnant in each group is given in

Table 9.3. Is there any evidence for believing that the training methods show

different proportions of pregnant animals?

null hypothesis : proportions pregnant are the same for the three methods. In the

same sense no association between the training methods and the pregnancy state

of the cows. The alternative hypothesis is that the proportions pregnant are not

equal.

Test statistic: Chi-squared test, With Yates’ correction

P-value

Insemination-Training

2 variables:

• Insemination Method (I, II, III)

• Pregnancy ( Preg, Non-preg)

Weight Cases

Weight Cases

The Same previous steps for Chi-square

(P = 0.008). we have evidence to reject the null hypothesis that the proportions

pregnant are the same for the three methods.

In this instance, Further analysis by comparing the proportions of pregnancies

obtained by any two methods shows that Method III has significantly lower

proportion of cows pregnant than Method I (test statistic = 8.66, P = 0.009 after

employing Bonferroni’s correction for multiple comparisons,

Chi-squared test for trend test the null hypothesis that there is no trend in the proportions.

2×C contingency table in which the variable defining the columns comprises c

ordered categories(e.g. body condition score or age categories). We are

interested in comparing the proportions of successes in c ordered groups, and

would expect any differences in the proportions, if they exist, to be related to the

ordering.

Comparing two paired proportions

McNemar’s test

Schönmann et al. (1994) compared two methods of culture of Tritrichomonas

foetus in the washings of the prepuce of infected beef bulls to determine the best

method for detection of the organism. In comparing the methods of culture,

Claussen’s medium detected the organism in 61 of 83 amples whereas a

commercial system detected the organism in 73 of the same 83 samples.

Trichomonas.sav

Variables 2; Claussen: 1;+Ve, 2-Ve

Commercial: 1;+Ve, 2-Ve

Null hypothesis :the true proportions detected are the same using Claussen’s

medium and the commercial system. The alternative hypothesis is that the two

proportions are different.

Test statistic:Chi-square

P-value

(P = 0.006).>>>reject the null hypothesis;>>>we conclude that the commercial

system has the ability to detect the greater proportion of organisms.

the proportions of organisms detected by Claussen’s medium and the

commercial system to be 61/83 = 0.735 and 73/83 = 0.880, respectively.