Categorical data analysis which part of the generalized linear model

Categorical Data Analysis
• Independent (Explanatory) Variable is
Categorical (Nominal or Ordinal)
• Dependent (Response) Variable is
Categorical (Nominal or Ordinal)
• Special Cases:
– 2x2 (Each variable has 2 levels)
– Nominal/Nominal
– Nominal/Ordinal
– Ordinal/Ordinal

Contingency Tables
• Tables representing all combinations of
levels of explanatory and response
variables
• Numbers in table represent Counts of the
number of cases in each cell
• Row and column totals are called
Marginal counts

Example – EMT Assessment of Kids
• Explanatory Variable
– Child Age (Infant,
Toddler, Pre-school,
School-age,
Adolescent)
• Response Variable –
EMT Assessment
(Accurate, Inaccurate)
Assessment
Age Acc Inac Tot
Inf 168 73 241
Tod 230 73 303
Pre 254 53 307
Sch 379 58 437
Ado 652 124 776
Tot 1683 381 2064
Source: Foltin, et al (2002)

2x2 Tables
• Each variable has 2 levels
– Explanatory Variable – Groups (Typically
based on demographics, exposure, or Trt)
– Response Variable – Outcome (Typically
presence or absence of a characteristic)
• Measures of association
– Relative Risk (Prospective Studies)
– Odds Ratio (Prospective or Retrospective)
– Absolute Risk (Prospective Studies)

2x2 Tables - Notation
Outcome
Present
Outcome
Absent
Group
Total
Group 1 n11 n12 n1.
Group 2 n21 n22 n2.
Outcome
Total
n.1 n.2 n..

Relative Risk
• Ratio of the probability that the outcome
characteristic is present for one group,
relative to the other
• Sample proportions with characteristic from
groups 1 and 2:
.
2
21
2
^
.
1
11
1
^
n
n
n
n

 


Relative Risk
• Estimated Relative Risk:
2
^
1
^



RR
95% Confidence Interval for Population Relative Risk:
21
2
^
11
1
^
96
.
1
96
.
1
)
1
(
)
1
(
71828
.
2
)
)
(
,
)
(
(
n
n
v
e
e
RR
e
RR v
v

 






Relative Risk
• Interpretation
– Conclude that the probability that the outcome
is present is higher (in the population) for
group 1 if the entire interval is above 1
is present is lower (in the population) for
group 1 if the entire interval is below 1
– Do not conclude that the probability of the
outcome differs for the two groups if the
interval contains 1

Example - Coccidioidomycosis and
TNF-antagonists
• Research Question: Risk of developing Coccidioidmycosis
associated with arthritis therapy?
• Groups: Patients receiving tumor necrosis factor  (TNF)
versus Patients not receiving TNF (all patients arthritic)
COC No COC Total
TNFa 7 240 247
Other 4 734 738
Total 11 974 985
Source: Bergstrom, et al (2004)

TNF-antagonists
• Group 1: Patients on TNF
• Group 2: Patients not on TNF
)
76
.
17
,
55
.
1
(
)
24
.
5
,
24
.
5
(
:
%
95
3874
.
4
0054
.
1
7
0283
.
1
24
.
5
0054
.
0283
.
0054
.
738
4
0283
.
247
7
3874
.
96
.
1
3874
.
96
.
1
2
^
1
^
2
^
1
^














e
e
CI
v
RR




Entire CI above 1  Conclude higher risk if on TNF

Odds Ratio
• Odds of an event is the probability it occurs
divided by the probability it does not occur
• Odds ratio is the odds of the event for group 1
divided by the odds of the event for group 2
• Sample odds of the outcome for each group:
22
21
2
12
11
.
1
12
.
1
11
1
/
/
n
n
odds
n
n
n
n
n
n
odds




Odds Ratio
• Estimated Odds Ratio:
21
12
22
11
22
21
12
11
2
1
/
/
n
n
n
n
n
n
n
n
odds
odds
OR 


95% Confidence Interval for Population Odds Ratio
22
21
12
11
96
.
1
96
.
1
1
1
1
1
71828
.
2
)
)
(
,
)
(
(
n
n
n
n
v
e
e
OR
e
OR v
v







Odds Ratio
• Interpretation
group 1 if the entire interval is above 1
group 1 if the entire interval is below 1
interval contains 1

Example - NSAIDs and GBM
• Case-Control Study (Retrospective)
– Cases: 137 Self-Reporting Patients with Glioblastoma
Multiforme (GBM)
– Controls: 401 Population-Based Individuals matched to
cases wrt demographic factors
GBMPresent GBMAbsent Total
NSAIDUser 32 138 170
NSAIDNon-User 105 263 368
Total 137 401 538
Source: Sivak-Sears, et al (2004)

Example - NSAIDs and GBM
)
91
.
0
,
37
.
0
(
)
58
.
0
,
58
.
0
(
:
%
95
0518
.
0
263
1
105
1
138
1
32
1
58
.
0
14490
8416
)
105
(
138
)
263
(
32
0518
.
0
96
.
1
0518
.
0
96
.
1










e
e
CI
v
OR
Interval is entirely below 1, NSAID use appears
to be lower among cases than controls

Absolute Risk
• Difference Between Proportions of outcomes
with an outcome characteristic for 2 groups
• Sample proportions with characteristic
from groups 1 and 2:
.
2
21
2
^
.
1
11
1
^
n
n
n
n

 


Absolute Risk
2
^
1
^

 

AR
Estimated Absolute Risk:
95% Confidence Interval for Population Absolute Risk
.
2
2
^
2
^
.
1
1
^
1
^
1
1
96
.
1
n
n
AR





















Absolute Risk
• Interpretation
group 1 if the entire interval is positive
group 1 if the entire interval is negative
interval contains 0

TNF-antagonists
• Group 1: Patients on TNF
• Group 2: Patients not on TNF
)
0242
.
0
,
0016
.
0
(
0213
.
0229
.
738
)
9946
(.
0054
.
247
)
9717
(.
0283
.
96
.
1
0229
.
:
%
95
0229
.
0054
.
0283
.
0054
.
738
4
0283
.
247
7
2
^
1
^
2
^
1
^














CI
AR 



Interval is entirely positive, TNF is
associated with higher risk

Fisher’s Exact Test
• Method of testing for association for 2x2 tables
when one or both of the group sample sizes is
small
• Measures (conditional on the group sizes and
number of cases with and without the
characteristic) the chances we would see
differences of this magnitude or larger in the
sample proportions, if there were no differences
in the populations

Example – Echinacea Purpurea for Colds
• Healthy adults randomized to receive EP
(n1.=24) or placebo (n2.=22, two were dropped)
• Among EP subjects, 14 of 24 developed cold
after exposure to RV-39 (58%)
• Among Placebo subjects, 18 of 22 developed
cold after exposure to RV-39 (82%)
• Out of a total of 46 subjects, 32 developed cold
• Out of a total of 46 subjects, 24 received EP
Source: Sperber, et al (2004)

Example – Echinacea Purpurea for Colds
• Conditional on 32 people
developing colds and 24
receiving EP, the
following table gives the
outcomes that would
have been as strong or
stronger evidence that EP
reduced risk of
developing cold (1-sided
test). P-value from SPSS
is .079.
EP/Cold Plac/Cold
14 18
13 19
12 20
11 21
10 22

Example - SPSS Output
Chi-Square Tests
2.990b 1 .084
1.984 1 .159
3.071 1 .080
.114 .079
46
Pearson Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
N of Valid Cases
Value df
Asymp. Sig.
(2-sided)
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
Computed only for a 2x2 table
a.
0 cells (.0%) have expected count less than 5. The minimum expected count is
6.70.
b.
TRT * COLD Crosstabulation
Count
10 14 24
4 18 22
14 32 46
EP
Placebo
TRT
Total
No Yes
COLD
Total

McNemar’s Test for Paired Samples
• Common subjects being observed under 2
conditions (2 treatments, before/after, 2
diagnostic tests) in a crossover setting
• Two possible outcomes (Presence/Absence of
Characteristic) on each measurement
• Four possibilities for each subjects wrt outcome:
– Present in both conditions
– Absent in both conditions
– Present in Condition 1, Absent in Condition 2
– Absent in Condition 1, Present in Condition 2

Condition 12 Present Absent
Present n11 n12
Absent n21 n22

• H0: Probability the outcome is Present is same
for the 2 conditions
• HA: Probabilities differ for the 2 conditions (Can
also be conducted as 1-sided test)
|)
|
(
2
)
05
.
0
96
.
1
(
|
|
:
.
.
:
.
.
2
/
21
12
21
12
obs
obs
obs
z
Z
P
val
P
if
z
z
R
R
n
n
n
n
z
S
T











Example - Reporting of Silicone Breast
Implant Leakage in Revision Surgery
• Subjects - 165 women having revision surgery
involving silicone gel breast implants
• Conditions (Each being observed on all women)
– Self Report of Presence/Absence of Rupture/Leak
– Surgical Record of Presence/Absence of Rupture/Leak
SELF * SURGICAL Crosstabulation
Count
69 28 97
5 63 68
74 91 165
Rupture
No Rupture
SELF
Total
Rupture No Rupture
SURGICAL
Total
Source: Brown and Pennello (2002)

Example - Reporting of Silicone Breast
Implant Leakage in Revision Surgery
• H0: Tendency to report ruptures/leaks is the
same for self reports and surgical records
• HA: Tendencies differ
0
|)
|
(
2
96
.
1
|
|
:
.
.
00
.
4
5
28
5
28
:
.
.
21
12
21
12












obs
obs
obs
z
Z
P
val
P
z
R
R
n
n
n
n
z
S
T

Pearson’s Chi-Square Test
• Can be used for nominal or ordinal explanatory
and response variables
• Variables can have any number of distinct levels
• Tests whether the distribution of the response
variable is the same for each level of the
explanatory variable (H0: No association
between the variables
• r = # of levels of explanatory variable
• c = # of levels of response variable

• Intuition behind test statistic
– Obtain marginal distribution of outcomes for
the response variable
– Apply this common distribution to all levels of
the explanatory variable, by multiplying each
proportion by the corresponding sample size
– Measure the difference between actual cell
counts and the expected cell counts in the
previous step

• Notation to obtain test statistic
– Rows represent explanatory variable (r levels)
– Cols represent response variable (c levels)
1 2 … c Total
1 n11
n12
… n1c
n1.
2 n21
n22
… n2c
n2.
… … … … … …
r nr1
nr2
… nrc
nr.
Total n.1
n.2
… n.c
n..

• Marginal distribution of response and expected cell
counts under hypothesis of no association:
..
.
.
^
.
..
.
^
..
1
.
1
^
)
(
n
n
n
n
n
E
n
n
n
n
j
i
j
i
ij
c
c






 

• H0: No association between variables
• HA: Variables are associated
)
(
:
.
.
)
(
))
(
(
:
.
.
2
2
2
)
1
)(
1
(
,
2
2
2
X
P
val
P
X
R
R
n
E
n
E
n
X
S
T
c
r
i j ij
ij
ij















Assessment
Age Acc Inac Tot
Inf 168 73 241
Tod 230 73 303
Pre 254 53 307
Sch 379 58 437
Ado 652 124 776
Tot 1683 381 2064
Assessment
Age Acc Inac Tot
Inf 197 44 241
Tod 247 56 303
Pre 250 57 307
Sch 356 81 437
Ado 633 143 776
Tot 1683 381 2064
Observed Expected

• Note that each expected count is the row total
times the column total, divided by the overall
total. For the first cell in the table:
197
2064
)
1683
(
241
)
(
..
1
.
.
1
11 


n
n
n
n
E
• The contribution to the test statistic for this cell is
27
.
4
197
)
197
168
( 2



• H0: No association between variables
• HA: Variables are associated
488
.
9
:
.
.
1
.
40
143
)
143
124
(
197
)
197
168
(
:
.
.
2
4
,
05
.
2
)
1
2
)(
1
5
(
,
05
.
2
2
2
2












 

X
R
R
X
S
T 
Reject H0, conclude that the accuracy of
assessments differs among age groups

Example - SPSS Output
AGE * ASSESS Crosstabulation
168 73 241
196.5 44.5 241.0
230 73 303
247.1 55.9 303.0
254 53 307
250.3 56.7 307.0
379 58 437
356.3 80.7 437.0
652 124 776
632.8 143.2 776.0
1683 381 2064
1683.0 381.0 2064.0
Count
Expected Count
Count
Expected Count
Count
Expected Count
Count
Expected Count
Count
Expected Count
Count
Expected Count
Infant
Toddler
Pre-school
School age
Adolescent
AGE
Total
Accurate Inaccurate
ASSESS
Total
Chi-Square Tests
40.073a 4 .000
37.655 4 .000
29.586 1 .000
2064
Pearson Chi-Square
Likelihood Ratio
Linear-by-Linear
Association
N of Valid Cases
Value df
Asymp. Sig.
(2-sided)
0 cells (.0%) have expected count less than 5. The
minimum expected count is 44.49.
a.

Ordinal Explanatory and Response
Variables
• Pearson’s Chi-square test can be used to test
associations among ordinal variables, but more
powerful methods exist
• When theories exist that the association is
directional (positive or negative), measures exist
to describe and test for these specific
alternatives from independence:
– Gamma
– Kendall’s b

Concordant and Discordant Pairs
• Concordant Pairs - Pairs of individuals where
one individual scores “higher” on both ordered
variables than the other individual
• Discordant Pairs - Pairs of individuals where one
individual scores “higher” on one ordered
variable and the other individual scores “higher”
on the other
• C = # Concordant Pairs D = # Discordant Pairs
– Under Positive association, expect C > D
– Under Negative association, expect C < D
– Under No association, expect C  D

Example - Alcohol Use and Sick Days
• Alcohol Risk (Without Risk, Hardly any Risk,
Some to Considerable Risk)
• Sick Days (0, 1-6, 7)
• Concordant Pairs - Pairs of respondents where
one scores higher on both alcohol risk and sick
days than the other
• Discordant Pairs - Pairs of respondents where
one scores higher on alcohol risk and the other
scores higher on sick days
Source: Hermansson, et al (2003)

ALCOHOL * SICKDAYS Crosstabulation
Count
347 113 145 605
154 63 56 273
52 25 34 111
553 201 235 989
Without Risk
Hardly any Risk
Some-Considerable Risk
ALCOHOL
Total
0 days 1-6 days 7+ days
SICKDAYS
Total
• Concordant Pairs: Each individual in a given cell
is concordant with each individual in cells
“Southeast” of theirs
•Discordant Pairs: Each individual in a given cell is
discordant with each individual in cells “Southwest”
of theirs

ALCOHOL * SICKDAYS Crosstabulation
Count
347 113 145 605
154 63 56 273
52 25 34 111
553 201 235 989
Without Risk
Hardly any Risk
Some-Considerable Risk
ALCOHOL
Total
0 days 1-6 days 7+ days
SICKDAYS
Total
73496
)
52
(
63
)
25
52
(
56
)
52
154
(
113
)
25
52
63
154
(
145
83164
)
34
(
63
)
34
25
(
154
)
34
56
(
113
)
34
25
56
63
(
347




















D
C

Measures of Association
• Goodman and Kruskal’s Gamma:
1
1
^
^






 

D
C
D
C
• Kendall’s b:
)
)(
(
2
.
2
2
.
2
^

 



j
i
b
n
n
n
n
D
C

When there’s no association between the ordinal variables,
the population based values of these measures are 0.
Statistical software packages provide these tests.

0617
.
0
73496
83164
73496
83164
^







D
C
D
C

Symmetric Measures
.035 .030 1.187 .235
.062 .052 1.187 .235
989
Kendall's tau-b
Gamma
Ordinal by
Ordinal
N of Valid Cases
Value
Asymp.
Std. Error
a
Approx. T
b
Approx. Sig.
Not assuming the null hypothesis.
a.
Using the asymptotic standard error assuming the null hypothesis.
b.

Categorical data analysis which part of the generalized linear model

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