From last time….

Basic Biostats Topics • Summary Statistics

– mean, median, mode– standard deviation, standard error

• Confidence Intervals

• Hypothesis Tests– t-test (paired and unpaired)– Chi-Square test– Fisher’s exact test

More Advanced

• Linear Regression

• Logistic Regression

• Repeated Measures Analysis

• Survival Analysis

• Analyzing fMRI data

General Biostatistics References• Practical Statistics for Medical Research.

Altman. Chapman and Hall, 1991.• Medical Statistics: A Common Sense Approach.

Campbell and Machin. Wiley, 1993• Principles of Biostatistics. Pagano and Gauvreau.

Duxbury Press, 1993.• Fundamentals of Biostatistics. Rosner. Duxbury

Press, 1993.

Lecture 3:Linear Regression

Elizabeth Garrett

esg@jhu.edu

Child Psychiatry Research Methods Lecture Series

Introduction

• Simple linear regression is most useful for looking at associations between continuous variables.

• We can evaluate if two variables are associated linearly.

• We can evaluate how well we can predict one of the variables if we know the other.

Motivating Example (Tierney et al. 2001)

• Is there an association between total sterol level and ADI scores in autistic children?

• Hypothesis: Children with lower sterol levels will tend to have poorer performance (i.e. higher scores) on the following components of the ADI:– social – nonverbal– repetitive

Preliminary Data

• 9 individuals with autism

• Some have been on cholesterol supplementation (7 out of 9)

• Mean age: 14

• Age range: 8 - 32 years

• Sterol is a continuous variable

• ADI scores are continuous variables

Statistical Language• Need to choose what variable is the predicted

(Y) and which is the predictor (X).

• Y: outcome, dependent variable, endogenous variable

• X: covariate, predictor, regressor, explanatory variable, exogenous variable, independent variable.

• Our example?

Sterol Level

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Sterol Level

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core How can we conclude

if there is or is not anassociation betweensterol and the ADI scores?

One approach: Correlation

• Correlation is a measure of LINEAR association between two variables.

• It takes values from -1 to 1.• Often notated r or

r = 1 perfect positive correlation

r = -1 perfect negative correlation

r = 0 no correlation

x

y

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81

0

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-10

-8-6

-4-2

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01

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r = 0.95

r = 0.09

r = 0.77

r = -0.95

Correlation between ADI measures and Sterol

Sterol Level

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r = -0.70

r = 0.06

r = -0.85

Related to r: R2

• R2 = % of variation in Y explained by X.

• Example: – Correlation between nonverbal score and sterol is -0.85.– R2 is 0.852 = 0.73– 73% of the variation in nonverbal score is explained by sterol

• Gives a sense of the value of sterol in predicting nonverbal score

• Other examples– R2 between sterol and social is 0.49– R2 between sterol and repetitive is 0.004

Simple Linear Regression (SLR) Approach

(1) Fits “best” line to describe the association between Y and X (note: straight line)

(2) Line can be described by two numbers

- intercept

- slope

(3) By-product of regression: correlation measures how close points fall from the line.

(4) Why “simple”? Only one X variable.

Intercept = 24.8

Slope = -0.01

Sterol Level

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SLR answers two questions….

• Association? – Does nonverbal score tend to decrease on average

when sterol increases?– Is slope different than zero?

• Prediction?– Can we predict nonverbal score if we know sterol

level?– Is the correlation (or R2) high?

• You CAN have association with low correlation!

n o n verb a l stero l 0 1Equation of a line: 0: Intercept

0 is the estimated nonverbal score if it were possible to have a sterol level of 0 (nonsensical in this case).

0 calibrates height of line

• 1: Slope 1 is the estimated change in nonverbal score for a one unit

change in sterol 1 the estimated difference in nonverbal score comparing two

kids whose sterol levels differ by one.

– We usually use 1 as our measure of association

The slope, 1

Is 1 different than zero?

Are each of these reasonable given the data that we have observed?

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Evaluating Association 1is a “statistic,” similar to a sample mean, and as such

has a precision estimate.

• The precision estimate is called the standard error of 1. Denoted se(1).

• We look at how large 1 is compared to its standard error

1 is often called a “regression coefficient” or a “slope.”

General Rule• If , then we say that 1 is

statistically significantly different than zero.

• T-test interpretation: H0: 1 = 0

Ha: 1 0

• If is true, then p-value less than 0.05.

• Intuition: 1 is large compared to its precision not likely that 1 is 0.

1

1

2se( )

1

1

2se( )

For large samples….

1/se(1) pvalue

0.50 0.62

1.00 0.32

1.50 0.13

1.65 0.10

1.96 0.050

2.00 0.047

2.25 0.025

2.58 0.010

2.75 0.006

3.00 0.003

3.25 0.001

ADI Nonverbal and Sterol

------------------------------------------------------------------------------

nonvrb | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

totster | -.0099066 .0022804 -4.344 0.003 -.0152988 -.0045144

_cons | 24.84349 2.578369 9.635 0.000 18.74661 30.94036

------------------------------------------------------------------------------

1

0

se(1)

1

1se( )

pvalue

R-squared = 0.73

Outcome

Predictor

Interpretation“Comparing two autistic kids whose sterol levels differ by 1,

we estimate that the one with lower sterol will have an ADI nonverbal score that is higher by 0.01 points.”

Put it in “real” units:

“Comparing two autistic kids whose sterol levels differ by 200, we estimate that the child with the lower sterol level will have an ADI nonverbal score that is higher by 2 points.”

(Note: 200 x 0.01 = 2.0)

A few other details...

• 95% Confidence interval interpretation: 1 2se(1) does not include

zero.

1/se(1) is called the – “t-statistic”

– “Z-statistic”

• If you have small sample (i.e. fewer than 50 individuals), need to use a “t-correction.”

N 1/se(1) pvalue

10 2.31 0.05

15 2.16 0.05

20 2.10 0.05

30 2.05 0.05

40 2.02 0.05

50 2.01 0.05

Relationship between correlation and SLR

Testing that correlation is equal to zero is equivalent to testing that the slope is equal to zero.

Can have strong association and low correlation

x

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r = 0.931 = 1.86pvalue < 0.001

r = 0.551 = 1.88pvalue < 0.001

Additional Points

(1) Association measured is LINEAR

x

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r = 0.02

Additional Points

(2) Difference (i.e. distance) between observed data and fitted line is called a residual, .

1. 0.74 2. -0.95 3. -2.53 4. 3.01 5. 2.52 6. 0.45 7. -3.15 8. -0.07 9. 0.59

. Sterol Level

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Additional Points

(3) Often see model equation as

n o n verb a l stero l 0 1

n o n verb a l stero l i Ni i i 0 1 1; , .. . ,

Generically,

y x i Ni i i 0 1 1; , .. . ,

n o n verb a l stero l 0 1

Refers toregressionlineRefer to

observeddata

Additional Points

(4) Spread of points around line is assumed to be constant (i.e. variance of residuals is constant)

x

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-20

02

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0

BAD!

Multiple Linear Regression

• More than one X variable

• Generally the same, except– Can’t make plots in multi-dimensions– Interpretation of ’s is somewhat different

y x x xi i i i i 0 1 1 2 2 3 3

Other ADI and Sterol SLRs

• How is age when supplementation began related to sterol?

• How is age when supplementation began related to nonverbal score?

nonvrb

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totster

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agester

How might this change our previous result?

Sterol Nonverbal Score

• What if age when cholesterol supplementation began is associated with both sterol level and nonverbal score?

• Is it correct to conclude that total sterol level is associated with nonverbal score?

Supplementation Age

We can “adjust”!

------------------------------------------------------------------------------

nonvrb | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

sterol | -.0105816 .0022118 -4.784 0.003 -.0159937 -.0051696

agester | .1570626 .1158509 1.356 0.224 -.1264143 .4405394

_cons | 23.81569 2.551853 9.333 0.000 17.57153 30.05985

------------------------------------------------------------------------------

n o n verb a l stero l a g esteri i i i 0 1 2

Interpretation of Betas• Now that we have “adjusted” for age at

supplementation, we need to include that in our result:“Comparing two kids who began cholesterol supplementation at the

same age and whose sterol levels differ by 250 units, we estimate that the child with the lower sterol level will have an ADI nonverbal score higher by 2 points.”

“Adjusting for age at supplementation, comparing two kids whose sterol levels differ by 250 units, we estimate…”

“Controlling for age at supplementation …..”

“Holding age at supplementation constant…..”

Collinearity

• If two variables are – correlated with each other– correlated with the outcome

• Then, when combined in a MLR model, it could happen that– neither is significant– only one is significant– both remain significant

ADI and Sterol

We say that cholesterol time and sterol are “collinear.”

Correlation Matrix

| nonvrb sterol agester

---------+---------------------------

nonvrb | 1.0000

sterol | -0.8541 1.0000

agester | 0.0531 0.2251 1.0000

Summing up example….

• After adjusting for age at supplementation, it appears that sterol is still a significant predictor of ADI nonverbal score.

• BUT!– Only NINE observations! With more, we would almost

CERTAINLY see even stronger associations!

– We haven’t controlled for other potential confounders:

• length of time on supplementation

• nonverbal score prior to supplementation