Categorical Data Analysis: Contingency Tables and Test of Independence - Prof. Michae Hudg, Study notes of Data Analysis & Statistical Methods

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Categorical Data: Contingency Tables Bios 662

Michael G. Hudgens, Ph.D. [email protected] http://www.bios.unc.edu/∼mhudgens 2006-10-17 17:

Contingency Tables

• Two-way (r × c) contingency table:

j i 1 2 · · · c 1 n 11 n 12 · · · n 1 c 2 n 21 n 22 · · · n 2 c ... ... ... ... ... r nr 1 nr 2 · · · nrc

• Notation:

ni· =

∑^ c

j=

nij n·j =

∑^ r

i=

nij

Contingency Table: Example

• A survey of physicians asked about the size of com- munity in which they were reared and the size of the community in which they practice Practice Reared <5k 5-49k 50-99k 100k+ Total <5k 40 38 32 37 147 5-49k 26 42 35 33 136 50-99k 24 26 34 31 115 100k+ 30 39 53 60 182 120 145 154 161 580

Contingency Table: Example

• A case-control study was conducted to investigate the relationship between age at first birth and breast cancer Age at 1st birth <20 20-24 25-29 30-34 ≥ 35 Total Case 320 1206 1011 463 220 3220 Control 1422 4432 2893 1092 406 10245 1742 5638 3904 1555 626 13465

Contingency Tables

• Breast cancer example H 0 : distribution across ages is the same for cases and controls H 0 : πij = πi′j; j = 1, 2 ,... , c
• Test of homogeneity/association

Test of Independence or Association

• Under H 0 , the expected frequency in the (i, j) cell is

Eij =

ni·n·j N

• Consider breast cancer example
  • If H 0 is true, would expect the proportion of women < 20 to be πˆ· 1 = n^11 N+^ n^21 = n N·^1
  • There are n 1 · cases, so we would expect E 11 = n 1 ·^ n N·^1 = n^1 N·n·^1 cases to be < 20 years old

Test of Independence

• Under H 0 , X^2 ∼ χ^2 (r−1)(c−1)
• Physician’s Example:

(r − 1)(c − 1) = 3 × 3 = 9

C. 05 = {X^2 : X^2 > χ^2. 95 , 9 = 16. 92 }

Physician’s Example

• Expected values

Practice Reared <5k 5-49k 50-99k 100k+ Total <5k 30.4 36.8 39.0 40.8 147 5-49k 28.1 34.0 36.1 37.8 136 50-99k 23.8 28.8 30.5 31.9 115 100k+ 37.7 45.5 48.3 50.5 182 120 145 154 161 580

Breast Cancer Example

• Underlying probabilities

Age at 1st birth <20 20-24 25-29 30-34 ≥ 35 Total Case π 11 π 12 π 13 π 14 π 15 1 Control π 21 π 22 π 23 π 24 π 25 1

Breast Cancer Example

• Null hypothesis

H 0 : π 1 j = π 2 j for j = 1, 2 , 3 , 4 , 5

• Can use same statistic

X^2 =

∑^2

i=

∑^ c

j=

(Oij − Eij)^2 Eij^ ∼^ χ

2 (c−1)

Breast Cancer Example

• Test statistic

X^2 = (320^ −^416 .6)

2

416. 6 +^ · · ·^ +

(406 − 476 .3)^2

476. 3 = 130.^3

• Rejection region

C. 05 = {X^2 : X^2 > χ^2. 95 , 4 = 9. 49 }

• Reject H 0
• The age distributions are not the same

Asymptotic Approximation

• Note the χ^2 distribution for X^2 is an approximation
• The approximation works well for if Eij ≥ 5 for all i, j
• If Eij < 5, a generalization of Fisher’s exact test can be employed or categories combined

Test for Trend

• Consider a 2 × c
• The χ^2 test for homogeneity does not tell us how the probabilities differ
• Rather, just if they differ
• If the categories of the column variable are ordered, a more powerful test is possible

Test for Trend

• Suppose columns = exposure are ordered
• Rows = disease (yes/no)
• Interested in detecting alternatives where the probabil- ity of disease proportional to exposure
• I.e., looking for a monotonic dose-response type rela- tionship