Study design and sampling (and more on ANOVA/regression)

39 Pages·2007·0.19 MB·English

Study design and sampling
(and more on
ANOVA/regression)
Tron Anders Moger
7.10.2007
Recall: Could put data in a table as
this:
• Each type of test was given three times for
each type of subject
Group: Test type
Block: Subject type Profile fit Mindbender Psych Out
Poor Cell: 65 68 62 69 71 67 75 75 78
Fair 74 79 76 72 69 69 70 69 65
Good 64 72 65 68 73 75 78 82 80
Excellent 83 82 84 78 78 75 76 77 75
Testing different types of wheat in a
field
Interested in finding out if different types of wheat
yields different crops
Outcome: E.g. wheat in pounds per acre
Group
Wheat 1 Wheat 2 Wheat 3
IIIIIIIIIIIIIIIII IIIIIIIIIIIIIIII IIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIII IIIIIIIIIIIIIIII IIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIII IIIIIIIIIIIIIIII IIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIII IIIIIIIIIIIIIIII IIIIIIIIIIIIIIIII
Your field resembles an ANOVA data matrix!
One-way ANOVA: Testing if mean crop per acre is
different for different types of wheat!
More complex designs:
• Want to test different fertilizers also
Group:
Block: Wheat 1 Wheat 2 Wheat 3
Fertilizer 1 IIIIIIIII IIIIIIIIII IIIIIIIIII
Fertilizer 2 IIIIIIIII IIIIIIIIII IIIIIIIIII
Fertilizer 3 IIIIIIIII IIIIIIIIII IIIIIIIIII
Do different wheat types give different crops?
Do different fertilizers give different crops?
Two-way ANOVA!
Do e.g fertilizer 1 work better for wheat 1 than for wheat 2
and 3?
Is there interaction between wheat and fertilizer?
Two-way ANOVA with interaction!
Groups and blocks
• In the example: Arbitrary if we put wheat
type in group or block
• Equally interested in wheat and fertilizer
effects in the example
• Another example: Want to test 3 different
treatments for e.g. asthma
• Only interested in treatment effect (Group)
• Design a study for one-way ANOVA,
everyone’s happy
Are we happy?
• Is an asthma patient an asthma patient no matter
what?
• Different types of asthma patients could give
different treatment effects (this is serious for
pharma companies)
• If we do one-way ANOVA, we won’t find out!
• Specifically, if we sample patients at random,
might end up with 5 patients of the type that
responds badly, and 50 patients of the other types
• Results for the 5 patients will drown in the results
for the others, so we won’t even suspect that
something is wrong
• Blocking variable: Ensure that you sample e.g 30
patients of each type
Asthma: Two-way ANOVA
• Still only really interested in the treatment effect
• But, would like to control for the confounding
effect of type of patient
• Model for one-way ANOVA: X =µ+G +ε
ij i ij
• µ is total mean, G is group effect, ε is N(0,σ2)
i ij
• σ2 includes variation due to everything else,
including patient type
• Only effect we describe in the model, is the
treatment effect
Asthma: Two-way ANOVA cont’d
• Two-way ANOVA model: X =µ+G +B +I +ε
ijl i j ij ijl
• Describe both treatment effect (G ), patient type
i
effect (B ) and interaction (I )
i ij
• Remove variation due to patient type from σ2 (and
from it’s primary estimator, MSE)
• Means that σ2 <σ2
two-way one-way
• Recall: Test for treatment effect (G =0), compares
i
MSG to MSE (MSG/MSE~F-dist), reject if
sufficiently large
• Similar tests for the other effects, but based on
MSB and MSI
Asthma: Two-way ANOVA cont’d
• If there is a treatment effect, MSG will be a biased
estimator for σ2
• If there is a block effect, denominator MSE will be
smaller here than MSW for one-way ANOVA
• Value of test statistic will be larger!
• Easier to get significant effects! (More power)
• Also get more correct estimates for the group
means (because of the sampling)
• Similar to regression: The more significant
variables you include in your model, the greater
R2 becomes, and you get more correct estimates
for the regression coefficients
• R2 increases because σ2 decreases the more
variables you include
ANOVA and linear regression
• Regression: Split the distance from each data point
to the total mean into:
– 1. Distance from mean to regression line
– 2. Distance from regression line to data point
• Got sums of squares SSR, SSE and SST
• Used for estimation and measuring how close data
points were to regression line (R2)
• However; also used for an F-test on whether all
B =0 (From slide of detailed explanations of SPSS
i
output)
• This is ANOVA in linear regression!