Transcript Binary or categorical outcomes (proportions)

Tests for Binary/Categorical
outcomes
Binary or categorical outcomes
(proportions)
Are the observations correlated?
Outcome
Variable
Binary or
categorical
(e.g.
fracture,
yes/no)
independent
correlated
Alternative to the chisquare test if sparse
cells:
Chi-square test:
McNemar’s chi-square test:
Fisher’s exact test: compares
Conditional logistic
regression: multivariate
McNemar’s exact test:
compares proportions between
more than two groups
compares binary outcome between
correlated groups (e.g., before and
after)
Relative risks: odds ratios
or risk ratios
Logistic regression:
multivariate technique used
when outcome is binary; gives
multivariate-adjusted odds
ratios
regression technique for a binary
outcome when groups are
correlated (e.g., matched data)
GEE modeling: multivariate
regression technique for a binary
outcome when groups are
correlated (e.g., repeated measures)
proportions between independent
groups when there are sparse data
(some cells <5).
compares proportions between
correlated groups when there are
sparse data (some cells <5).
Binary or categorical outcomes
(proportions)
Are the observations correlated?
Outcome
Variable
Binary or
categorical
(e.g.
fracture,
yes/no)
independent
correlated
Alternative to the chisquare test if sparse
cells:
Chi-square test:
McNemar’s chi-square test:
Fisher’s exact test: compares
Conditional logistic
regression: multivariate
McNemar’s exact test:
compares proportions between
more than two groups
compares binary outcome between
correlated groups (e.g., before and
after)
Relative risks: odds ratios
or risk ratios (for 2x2 tables)
Logistic regression:
multivariate technique used
when outcome is binary; gives
multivariate-adjusted odds
ratios
regression technique for a binary
outcome when groups are
correlated (e.g., matched data)
GEE modeling: multivariate
regression technique for a binary
outcome when groups are
correlated (e.g., repeated measures)
proportions between independent
groups when there are sparse data
(some cells <5).
compares proportions between
correlated groups when there are
sparse data (some cells <5).
Chi-square test
From an RCT of probiotic supplementation during pregnancy to prevent
eczema in the infant:
Table 3. Cumulative incidence of eczema at 12 months of age
Cumulative incidence at
12 months
Probiotics group
Placebo group
p-value
Adjusted OR(95% CI)
p-value
12/33 (36.4%)
22/35 (62.9%)
0.029*
0.243(0.075–0.792)
0.019†
*Significant difference between the groups as determined by Pearson's chi-square test.
†p value was calculated by multivariable logistic regression analysis adjusted for the antibiotics use, total duration of breastfeeding,
and delivery by cesarean section.
Kim et al. Effect of probiotic mix (Bifidobacterium bifidum, Bifidobacterium lactis, Lactobacillus acidophilus) in the primary prevention of eczema: a double-blind,
randomized, placebo-controlled trial. Pediatric Allergy and Immunology. Published online October 2009.
Chi-square test
Statistical question: Does the proportion of infants with
eczema differ in the treatment and control groups?

What is the outcome variable? Eczema in the first
year of life (yes/no)

What type of variable is it? Binary

Are the observations correlated? No

Are groups being compared and, if so, how many?
Yes, two groups

Are any of the counts smaller than 5? No, smallest is
12 (probiotics group with eczema)
 chi-square test or relative risks, or both
Chi-square test of
Independence
Chi-square test allows you to compare proportions between 2
or more groups (ANOVA for means; chi-square for
proportions).
Example 2

Asch, S.E. (1955). Opinions and social
pressure. Scientific American, 193, 3135.
The Experiment



A Subject volunteers to participate in a
“visual perception study.”
Everyone else in the room is actually a
conspirator in the study (unbeknownst
to the Subject).
The “experimenter” reveals a pair of
cards…
The Task Cards
Standard line
Comparison lines
A, B, and C
The Experiment




Everyone goes around the room and says
which comparison line (A, B, or C) is correct;
the true Subject always answers last – after
hearing all the others’ answers.
The first few times, the 7 “conspirators” give
the correct answer.
Then, they start purposely giving the
(obviously) wrong answer.
75% of Subjects tested went along with the
group’s consensus at least once.
Further Results


In a further experiment, group size
(number of conspirators) was altered
from 2-10.
Does the group size alter the proportion
of subjects who conform?
The Chi-Square test
Number of group members?
Conformed?
2
4
6
8
10
Yes
20
50
75
60
30
No
80
50
25
40
70
Apparently, conformity less likely when less or more group
members…



20 + 50 + 75 + 60 + 30 = 235
conformed
out of 500 experiments.
Overall likelihood of conforming =
235/500 = .47
Expected frequencies if no
association between group
size and conformity…
Number of group members?
Conformed?
2
4
6
8
10
Yes
47
47
47
47
47
No
53
53
53
53
53

Do observed and expected differ more
than expected due to chance?
Chi-Square test
(observed - expected) 2
 
expected
2
(20  47) 2 (50  47) 2 (75  47) 2 (60  47) 2 (30  47) 2
4 





47
47
47
47
47
(80  53) 2 (50  53) 2 (25  53) 2 (40  53) 2 (70  53) 2




 85
53
53
53
53
53
2
Degrees of freedom = (rows-1)*(columns-1)=(2-1)*(5-1)=4
Chi-Square test
(observed - expected) 2
 
expected
2
(20  47) 2 (50  47) 2 (75  47) 2 (60  47) 2 (30  47) 2
4 





47
47
47
47
47
(80  53) 2 (50  53) 2 (25  53) 2 (40  53) 2 (70  53) 2




 85
53
53
53
53
53
2
Degrees of freedom = (rows-1)*(columns-1)=(2-1)*(5-1)=4
Rule of thumb: if the chi-square statistic is much greater than it’s degrees of freedom,
indicates statistical significance. Here 85>>4.
Interpretation



Group size and conformity are not
independent, for at least some categories of
group size
The proportion who conform differs between
at least two categories of group size
Global test (like ANOVA) doesn’t tell you
which categories of group size differ
Caveat
**When the sample size is very small in
any cell (<5), Fisher’s exact test is
used as an alternative to the chi-square
test.
Review Question 1
I divide my study population into smokers, ex-smokers,
and never-smokers; I want to compare years of
schooling (a normally distributed variable) between the
three groups. What test should I use?
a.
b.
c.
d.
e.
Repeated-measures ANOVA.
One-way ANOVA.
Difference in proportions test.
Paired ttest.
Chi-square test.
Review Question 1
I divide my study population into smokers, ex-smokers,
and never-smokers; I want to compare years of
schooling (a normally distributed variable) between the
three groups. What test should I use?
a.
b.
c.
d.
e.
Repeated-measures ANOVA.
One-way ANOVA.
Difference in proportions test.
Paired ttest.
Chi-square test.
Review Question 2
I divide my study population into smokers, ex-smokers,
and never-smokers; I want to compare the proportions
of each group that went to graduate school. What test
should I use?
a.
b.
c.
d.
e.
Repeated-measures ANOVA.
One-way ANOVA.
Difference in proportions test.
Paired ttest.
Chi-square test.
Review Question 2
I divide my study population into smokers, ex-smokers,
and never-smokers; I want to compare the proportions
of each group that went to graduate school. What test
should I use?
a.
b.
c.
d.
e.
Repeated-measures ANOVA.
One-way ANOVA.
Difference in proportions test.
Paired ttest.
Chi-square test.
Review Question 2
I divide my study population into smokers, ex-smokers,
and never-smokers; I want to compare the proportions
of each group that went to graduate school. What test
should I use?
a.
b.
c.
d.
e.
Repeated-measures ANOVA.
One-way ANOVA.
Difference in proportions test.
Paired ttest.
Chi-square test.
Binary or categorical outcomes
(proportions)
Are the observations correlated?
Outcome
Variable
Binary or
categorical
(e.g.
fracture,
yes/no)
independent
correlated
Alternative to the chisquare test if sparse
cells:
Chi-square test:
McNemar’s chi-square test:
Fisher’s exact test: compares
Conditional logistic
regression: multivariate
McNemar’s exact test:
compares proportions between
more than two groups
compares binary outcome between
correlated groups (e.g., before and
after)
Relative risks: odds ratios
or risk ratios (for 2x2 tables)
Logistic regression:
multivariate technique used
when outcome is binary; gives
multivariate-adjusted odds
ratios
regression technique for a binary
outcome when groups are
correlated (e.g., matched data)
GEE modeling: multivariate
regression technique for a binary
outcome when groups are
correlated (e.g., repeated measures)
proportions between independent
groups when there are sparse data
(some cells <5).
compares proportions between
correlated groups when there are
sparse data (some cells <5).
Risk ratios and odds ratios
From an RCT of probiotic supplementation during pregnancy to prevent
eczema in the infant:
Table 3. Cumulative incidence of eczema at 12 months of age
Cumulative incidence at
12 months
Probiotics group
Placebo group
p-value
Adjusted OR(95% CI)
p-value
12/33 (36.4%)
22/35 (62.9%)
0.029*
0.243(0.075–0.792)
0.019†
*Significant difference between the groups as determined by Pearson's chi-square test.
†p value was calculated by multivariable logistic regression analysis adjusted for the antibiotics use, total duration of breastfeeding,
and delivery by cesarean section.
Kim et al. Effect of probiotic mix (Bifidobacterium bifidum, Bifidobacterium lactis, Lactobacillus acidophilus) in the primary prevention of eczema: a double-blind,
randomized, placebo-controlled trial. Pediatric Allergy and Immunology. Published online October 2009.
Corresponding 2x2 table
Treatment Group
Treatment
Placebo
+
12
22
-
21
13
Eczema
Risk ratios and odds ratios
Statistical question: Does the proportion of infants with
eczema differ in the treatment and control groups?

What is the outcome variable? Eczema in the first
year of life (yes/no)

What type of variable is it? Binary

Are the observations correlated? No

Are groups being compared and, if so, how many?
Yes, binary

Are any of the counts smaller than 5? No, smallest is
12 (probiotics group with eczema)
 chi-square test or relative risks, or both
Odds vs. Risk (=probability)
If the risk is…
Then the odds
are…
½ (50%)
1:1
¾ (75%)
3:1
1/10 (10%)
1:9
1/100 (1%)
1:99
Note: An odds is always higher than its corresponding probability,
unless the probability is 100%.
Risk ratios and odds ratios



Absolute risk difference in eczema
between treatment and placebo:
36.4%-62.9%=-26.5% (p=.029, chisquare test).
There is a 26.5%
36.4%
 0.58 decrease in absolute risk,
Risk ratio:
a 42% decrease in relative
62.9%
risk, and a 66% decrease
Corresponding odds ratio:
in relative odds.
36.4% /(1  36.4%)
 0.34
62.9% /(1  62.9%)
Why do we ever use an odds
ratio??



We cannot calculate a risk ratio from a casecontrol study (since we cannot calculate the
risk of developing the disease in either
exposure group).
The multivariate regression model for binary
outcomes (logistic regression) gives odds
ratios, not risk ratios.
The odds ratio is a good approximation of the
risk ratio when the disease/outcome is rare
(~<10% of the control group)
Interpretation of the odds
ratio:


The odds ratio will always be bigger
than the corresponding risk ratio if RR
>1 and smaller if RR <1 (the harmful or
protective effect always appears larger)
The magnitude of the inflation depends
on the prevalence of the disease.
The rare disease assumption
OR 
P( D / E )
P (~ D / E ) 1
P( D / E )

P( D / ~ E )
P( D / ~ E )
P (~ D / ~ E )
 RR
1
When a disease is rare:
P(~D) = 1 - P(D)  1
The odds ratio vs. the risk ratio
Rare Outcome
Odds ratio
Odds ratio
Risk ratio
1.0 (null)
Risk ratio
Common Outcome
Odds ratio
Odds ratio
Risk ratio
1.0 (null)
Risk ratio
When is the OR is a good
approximation of the RR?
General Rule of
Thumb:
“OR is a good
approximation as long
as the probability of the
outcome in the
unexposed is less than
10%”
Binary or categorical outcomes
(proportions)
Are the observations correlated?
Outcome
Variable
Binary or
categorical
(e.g.
patency,
revision)
independent
correlated
Alternative to the chisquare test if sparse
cells:
Chi-square test:
McNemar’s chi-square test:
Fisher’s exact test: compares
Conditional logistic
regression: multivariate
McNemar’s exact test:
compares proportions between
more than two groups
compares binary outcome between
correlated groups (e.g., before and
after)
Relative risks: odds ratios
or risk ratios (for 2x2 tables)
Logistic regression:
multivariate technique used
when outcome is binary; gives
multivariate-adjusted odds
ratios
regression technique for a binary
outcome when groups are
correlated (e.g., matched data)
GEE modeling: multivariate
regression technique for a binary
outcome when groups are
correlated (e.g., repeated measures)
proportions between independent
groups when there are sparse data
(some cells <5).
compares proportions between
correlated groups when there are
sparse data (some cells <5).
Recall…

Split-face trial:



Researchers assigned 56 subjects to apply
SPF 85 sunscreen to one side of their faces
and SPF 50 to the other prior to engaging
in 5 hours of outdoor sports during midday.
Sides of the face were randomly assigned;
subjects were blinded to SPF strength.
Outcome: sunburn
Russak JE et al. JAAD 2010; 62: 348-349.
Results:
Table I -- Dermatologist grading of sunburn after an average of 5 hours of
skiing/snowboarding (P = .03; Fisher’s exact test)
Sun protection factor
Sunburned
Not sunburned
85
1
55
50
8
48
The authors use Fisher’s exact test to compare 1/56 versus 8/56. But this
counts individuals twice and ignores the correlations in the data!
McNemar’s test
Statistical question: Is SPF 85 more effective than SPF
50 at preventing sunburn?

What is the outcome variable? Sunburn on half a
face (yes/no)

What type of variable is it? Binary

Are the observations correlated? Yes, split-face trial

Are groups being compared and, if so, how many?
Yes, two groups (SPF 85 and SPF 50)

Are any of the counts smaller than 5? Yes, smallest
is 0
 McNemar’s test exact test (if bigger numbers, would
use McNemar’s chi-square test)
Correct analysis of data…
Table 1. Correct presentation of the data from: Russak JE et
al. JAAD 2010; 62: 348-349. (P = .016; McNemar’s test).
SPF-50 side
SPF-85 side
Sunburned
Not sunburned
Sunburned
1
0
Not sunburned
7
48
Only the 7 discordant pairs provide useful information for the analysis!
McNemar’s exact test…



There are 7 discordant pairs; under the null
hypothesis of no difference between
sunscreens, the chance that the sunburn
appears on the SPF 85 side is 50%.
In other words, we have a binomial
distribution with N=7 and p=.5.
What’s the probability of getting X=0 from a
binomial of N=7, p=.5?
7 7 0
 .5 .5  .0078
0

Probability =

Two-sided probability
=
7
  7 0
7 0 7
 .5 .5  .0078   .5 .5  .0078  .0156
0
7
McNemar’s chi-square test

Basically the same as McNemar’s exact
test but approximates the binomial
distribution with a normal distribution
(works well as long as sample sizes in
each cell >=5)
Binary or categorical outcomes
(proportions)
Are the observations correlated?
Outcome
Variable
Binary or
categorical
(e.g.
patency,
revision)
independent
correlated
Alternative to the chisquare test if sparse
cells:
Chi-square test:
McNemar’s test: compares
Fisher’s exact test: compares
Relative risks: odds ratios
Conditional logistic
regression: multivariate
McNemar’s exact test:
compares proportions between
more than two groups
or risk ratios (for 2x2 tables)
Logistic regression:
multivariate technique used
when outcome is binary; gives
multivariate-adjusted odds
ratios
binary outcome between correlated
groups (e.g., before and after)
regression technique for a binary
outcome when groups are
correlated (e.g., matched data)
GEE modeling: multivariate
regression technique for a binary
outcome when groups are
correlated (e.g., repeated measures)
proportions between independent
groups when there are sparse data
(some cells <5).
compares proportions between
correlated groups when there are
sparse data (some cells <5).
Recall: Political party and
drinking…
Drinking by political affiliation
Recall: Political party and
alcohol…
This association could be analyzed by a ttest
or a linear regression or also by logistic
regression:
Republican (yes/no) becomes the binary
outcome.
Alcohol (continuous) becomes the predictor.
Logistic regression






Statistical question: Does alcohol drinking
predict political party?
What is the outcome variable? Political party
What type of variable is it? Binary
Are the observations correlated? No
Are groups being compared? No, our
independent variable is continuous
 logistic regression
The logistic model…
ln(p/1- p)
=  + 1*X
Logit function
=log odds of the
outcome
The Logit Model (multivariate)
P ( D)
ln(
)    β1 ( X 1 )  β2 ( X 2 )...
1  P ( D)
Baseline odds
Logit function (log odds)
Linear function of
risk factors for
individual i:
1x1 + 2x2 + 3x3 +
4x4 …
Review question 7

a.
b.
c.
d.
e.
If X=.50, what is the logit (=log odds) of X?
.50
0
1.0
2.0
-.50
Review question 7

a.
b.
c.
d.
e.
If X=.50, what is the logit (=log odds) of X?
.50
0
1.0
2.0
-.50
Example: political party and
drinking…
Model:
Log odds of being a Republican (outcome)=
Intercept+ Weekly drinks (predictor)
Fit the data in logistic regression using a
computer…
Fitted logistic model:
“Log Odds” of being a Republican = -.09 -1.4* (d/wk)
Slope for
drinking can be
directly
translated into
an odds ratio:
e
1.4
 0.25
Interpretation: every 1 drink more per week decreases your odds of
being a Republican by 75% (95% CI is 0.047 to 1.325; p=.10)
To get back to OR’s…
P ( D)
ln(
)    β1 ( X 1 )  β2 ( X 2 )...
1  P ( D)
P( D)
  β1 ( X 1 )  β2 ( X 2 )...
 odds of disease 
e
1  P( D)
“Adjusted” Odds Ratio
Interpretation
odds of disease for the exposed
OR 
odds of disease for the unexposed

   alcohol(1)   smoking(1)
e
   alcohol( 0 )   smoking(1)
e
  alcohol(1)  smoking(1)
e e
e
   ( 0)  smoking(1) 
e e alcohol e
e
 alcohol(1)
1
e
 alcohol(1)
Adjusted odds ratio,
continuous predictor
odds of disease for the exposed
OR 
odds of disease for the unexposed
   alcohol(1)   smoking(1)   age( 29)
e
    alcohol(1)  smoking(1)  age(19)
e
  alcohol(1)  smoking(1)  age( 29)
 age( 29)
e e
e
e
e
   (1)  smoking(1)  age(19)   (19)  e  age(10)
age
e e alcohol e
e
e
Practical Interpretation
e
ˆ rf ( x )
 ORrisk factor of interest
The odds of disease increase multiplicatively by eß
for for every one-unit increase in the exposure,
controlling for other variables in the model.
Multivariate logistic regression
Litvick JR et al. Predictors of Olfactory Dysfunction in Patients With Chronic Rhinosinusitis. The Laryngoscope Dec 2008; 118: pp 2225-2230.
Logistic regression
Statistical question: What factors are associated with
anosmia (and hyposmia)?

What are the outcome variables? anosmia vs.
normal olfaction (and hyosmia vs. normal)

What type of variable is it? Binary

Are the observations correlated? No

Are groups being compared? We want to examine
multiple predictors at once, so we need multivariate
regression.
 multivariate logistic regression
Multivariate logistic regression
Interpretation:
being a smoker
increases your
odds of anosmia
by 658% after
adjusting for older
age, nasal
polyposis, asthma,
inferior turbinate
hypertrophy, and
septal deviation.
Litvick JR et al. Predictors of Olfactory Dysfunction in Patients With Chronic Rhinosinusitis. The Laryngoscope Dec 2008; 118: pp 2225-2230.
Logistic regression in crosssectional and cohort studies…

Many cohort and cross-sectional studies report ORs
rather than RRs even though the data necessary to
calculate RRs are available. Why?




If you have a binary outcome and want to adjust for
confounders, you have to use logistic regression.
Logistic regression gives adjusted odds ratios, not risk ratios.
These odds ratios must be interpreted cautiously (as
increased odds, not risk) when the outcome is common.
When the outcome is common, authors should also report
unadjusted risk ratios and/or use a simple formula to
convert adjusted odds ratios back to adjusted risk ratios.
Example, wrinkle study…

A cross-sectional study on risk factors for
wrinkles found that heavy smoking
significantly increases the risk of prominent
wrinkles.



Adjusted OR=3.92 (heavy smokers vs.
nonsmokers) calculated from logistic regression.
Interpretation: heavy smoking increases risk of
prominent wrinkles nearly 4-fold??
The prevalence of prominent wrinkles in nonsmokers is roughly 45%. So, it’s not possible to
have a 4-fold increase in risk (=180%)!
Raduan et al. J Eur Acad Dermatol Venereol. 2008 Jul 3.
Interpreting ORs when the
outcome is common…





If the outcome has a 10% prevalence in the
unexposed/reference group*, the maximum possible
RR=10.0.
For 20% prevalence, the maximum possible RR=5.0
For 30% prevalence, the maximum possible RR=3.3.
For 40% prevalence, maximum possible RR=2.5.
For 50% prevalence, maximum possible RR=2.0.
*Authors should report the prevalence/risk of the outcome in the
unexposed/reference group, but they often don’t. If this number is not given,
you can usually estimate it from other data in the paper (or, if it’s important
enough, email the authors).
Interpreting ORs when the
outcome is common…
If data are from a cross-sectional or cohort study, then you can
convert ORs (from logistic regression) back to RRs with a simple
formula:
OR
RR 
(1  Po )  ( Po  OR )
Where:
OR = odds ratio from logistic regression (e.g., 3.92)
P0 = P(D/~E) = probability/prevalence of the outcome in the
unexposed/reference group (e.g. ~45%)
Formula from: Zhang J. What's the Relative Risk? A Method of Correcting the Odds
Ratio in Cohort Studies of Common Outcomes JAMA. 1998;280:1690-1691.
For wrinkle study…
RRsmokers vs. nonsmokers
3.92

 1.69
(1  .45)  (.45  3.92)
So, the risk (prevalence) of wrinkles is increased by
69%, not 292%.
Zhang J. What's the Relative Risk? A Method of Correcting the Odds Ratio in Cohort
Studies of Common Outcomes JAMA. 1998;280:1690-1691.
Sleep and hypertension
study…





ORhypertension= 5.12 for chronic insomniacs who sleep
≤ 5 hours per night vs. the reference (good sleep)
group.
ORhypertension = 3.53 for chronic insomiacs who sleep
5-6 hours per night vs. the reference group.
Interpretation: risk of hypertension is increased
500% and 350% in these groups?
No, ~25% of reference group has hypertension. Use
formula to find corresponding RRs = 2.5, 2.2
Correct interpretation: Hypertension is increased
150% and 120% in these groups.
-Sainani KL, Schmajuk G, Liu V. A Caution on Interpreting Odds Ratios. SLEEP, Vol. 32, No. 8, 2009 .
-Vgontzas AN, Liao D, Bixler EO, Chrousos GP, Vela-Bueno A. Insomnia with objective short sleep duration is
associated with a high risk for hypertension. Sleep 2009;32:491-7.
Review problem 8

a.
b.
c.
d.
In a cross-sectional study of heart disease in middle-aged
men and women, 10% of men in the sample had
prevalent heart disease compared with only 5% of
women. After adjusting for age in multivariate logistic
regression, the odds ratio for heart disease comparing
males to females was 1.1 (95% confidence interval:
0.80—1.42). What conclusion can you draw?
Being male increases your risk of heart disease.
Age is a confounder of the relationship between gender and heart
disease.
There is a statistically significant association between gender and
heart disease.
The study had insufficient power to detect an effect.
Review problem 8

a.
b.
c.
d.
In a cross-sectional study of heart disease in middle-aged
men and women, 10% of men in the sample had
prevalent heart disease compared with only 5% of
women. After adjusting for age in multivariate logistic
regression, the odds ratio for heart disease comparing
males to females was 1.1 (95% confidence interval:
0.80—1.42). What conclusion can you draw?
Being male increases your risk of heart disease.
Age is a confounder of the relationship between gender and
heart disease.
There is a statistically significant association between gender and
heart disease.
The study had insufficient power to detect an effect.
Review topic: Diagnostic
Testing and Screening Tests
Characteristics of a diagnostic test
Sensitivity= Probability that, if you truly
have the disease, the diagnostic test
will catch it.
Specificity=Probability that, if you truly do
not have the disease, the test will
register negative.
Calculating sensitivity and
specificity from a 2x2 table
Screening Test
+
-
+
a
b
a+b
-
c
d
c+d
Truly have disease
a Among those with true
Sensitivity 
disease, how many test
a  b positive?
d
Specificity 
cd
Among those without the
disease, how many test
negative?
Hypothetical Example
Mammography
+
-
+
9
1
10
-
109
881
990
Breast cancer ( on biopsy)
Sensitivity=9/10=.90
1 false negatives out of 10
cases
Specificity= 881/990 =.89
109 false positives out of 990
Positive predictive value


The probability that if you test positive
for the disease, you actually have the
disease.
Depends on the characteristics of the
test (sensitivity, specificity) and the
prevalence of disease.
Calculating PPV and NPV from
a 2x2 table
Screening Test
+
-
+
a
b
-
c
d
Truly have disease
a+c
PPV
a

ac
b+d
Among those who test
positive, how many truly have
the disease?
NPV
d

bd
Among those who test
negative, how many truly do
not have the disease?
Hypothetical Example
Mammography
+
-
+
9
1
-
109
881
118
882
Breast cancer ( on biopsy)
PPV=9/118=7.6%
NPV=881/882=99.9%
Prevalence of disease = 10/1000 =1%
What if disease was twice as
prevalent in the population?
Mammography
+
-
+
18
2
20
-
108
872
980
Breast cancer ( on biopsy)
sensitivity=18/20=.90
specificity=872/980=.89
Sensitivity and specificity are characteristics of the test, so they don’t
change!
What if disease was more
prevalent?
Mammography
+
-
+
18
2
-
108
872
126
874
Breast cancer ( on biopsy)
PPV=18/126=14.3%
NPV=872/874=99.8%
Prevalence of disease = 20/1000 =2%
Conclusions


Positive predictive value increases with
increasing prevalence of disease
Or if you change the diagnostic tests to
improve their accuracy.
Review question 9
In a group of patients presenting to the hospital casualty department
with abdominal pain, 30% of patients have acute appendicitis. 70% of
patients with appendicitis have a temperature greater than 37.5ºC; 40%
of patients without appendicitis have a temperature greater than 37.5ºC.
a.
b.
c.
d.
e.
The sensitivity of temperature greater than 37.5ºC as a marker for
appendicitis is 21/49.
The specificity of temperature greater than 37.5ºC as a marker for
appendicitis is 42/70.
The positive predictive value of temperature greater than 37.5ºC as a
marker for appendicitis is 21/30.
The predictive value of the test will be the same in a different population.
The specificity of the test will depend upon the prevalence of appendicitis
in the population to which it is applied.
Review question 9
In a group of patients presenting to the hospital casualty department
with abdominal pain, 30% of patients have acute appendicitis. 70% of
patients with appendicitis have a temperature greater than 37.5ºC; 40%
of patients without appendicitis have a temperature greater than 37.5ºC.
a.
b.
c.
d.
e.
The sensitivity of temperature greater than 37.5ºC as a marker for
appendicitis is 21/49.
The specificity of temperature greater than 37.5ºC as a marker
for appendicitis is 42/70.
The positive predictive value of temperature greater than 37.5ºC as a
marker for appendicitis is 21/30.
The predictive value of the test will be the same in a different population.
The specificity of the test will depend upon the prevalence of appendicitis
in the population to which it is applied.