Randomized
Clinical Trials
(RCT)
Randomized
Clinical Trials
(RCT)
Designed to compare two or more treatment
groups for a statistically significant difference
between them – i.e., beyond random chance –
often measured via a “p-value” (e.g., p < .05).
Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx
 Let X = decrease (–) in cholesterol level (mg/dL); possible expected distributions:
Experiment
Treatment
population
Control
population
significant?
1
2
H 0 : 1  2
X
0
Patients
satisfying
inclusion
criteria
R
A
N
D
O
M
I
Z
E
Treatment
Arm
End of Study
RANDOM
SAMPLES
Control
Arm
T-test
F-test
(ANOVA)
Randomized
Clinical Trials
(RCT)
Designed to compare two or more treatment
groups for a statistically significant difference
between them – i.e., beyond random chance –
often measured via a “p-value” (e.g., p < .05).
Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx
 Let T = Survival time (months); population survival curves:
S(t) = P(T > t)
1
survival
probability
Kaplan-Meier
estimates
AUC difference
significant?
S2(t)
Control
S1(t)
Treatment
0
H 0 : S1 (t )  S2 (t )
T
End of Study
Log-Rank Test,
Cox Proportional
Hazards Model
Case-Control
studies
Cohort
studies
Observational study designs that test for a statistically significant association
between a disease D and exposure E to a potential risk (or protective) factor,
measured via “odds ratio,” “relative risk,” etc. Lung cancer / Smoking
Case-Control
studies
Cohort
studies
PRESENT
PAST
FUTURE
cases
E+ vs. E– ?
controls
reference group
D+ vs. D– E+ vs. E–
 relatively easy and inexpensive
subject to faulty records, “recall bias”
D+ vs. D– ?
 measures direct effect of E on D
expensive, extremely lengthy…
Example: Framingham, MA study
Both types of study yield a 22 “contingency table” of data:
D+
D–
E+
a
b
a+b
E–
c
d
c+d
a+c b+d
n
where a, b, c, d are the
numbers of individuals
in each cell.
H0: No association
between D and E.
End of Study
Chi-squared Test
McNemar Test
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Y
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
JAMA. 2003;290:1486-1493
Scatterplot
measures the strength
of linear association
between X and Y
r
–1
X
0
negative linear
correlation
+1
positive linear
correlation
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Y
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
JAMA. 2003;290:1486-1493
Scatterplot
measures the strength
of linear association
between X and Y
r
–1
X
0
negative linear
correlation
+1
positive linear
correlation
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Y
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
JAMA. 2003;290:1486-1493
Scatterplot
measures the strength
of linear association
between X and Y
r
–1
X
0
negative linear
correlation
+1
positive linear
correlation
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
measures the strength
of linear association
between X and Y
For this example, r = –0.387
(weak, negative linear correl)
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
residuals
Simple Linear Regression
gives the “best” line
that fits the data.
Regression
Methods
Want the unique line that minimizes
the sum of the squared residuals.
For this example, r = –0.387
(weak, negative linear correl)
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
residuals
Simple Linear Regression
gives the “least squares”
regression line.
Regression
Methods
Want the unique line that minimizes
the sum of the squared residuals.
For this example, r = –0.387
(weak, negative linear correl)
Y = 8.790 – 4.733 X (p = .0055)
Furthermore however, the proportion of total
variability in the data that is accounted for by
the line is only r 2 = (–0.387)2 = 0.1497 (15%).
• Polynomial Regression – predictors X, X2, X3,…
• Multilinear Regression – independent predictors X1, X2,…
w/o or w/ interaction (e.g., X5 X8)
• Logistic Regression – binary response Y (= 0 or 1)
• Transformations of data, e.g., semi-log, log-log,…
• Generalized Linear Models
• Nonlinear Models
• many more…
Sincere thanks to
• Judith Payne
• Heidi Miller
• Rebecca Mataya
• Troy Lawrence
• YOU!