biol 582
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BIOL 582. Lecture Set 6 Tests for multiple groups: Part 2 One-Way ANOVA via the General Linear Model (GLM). BIOL 582. Intro to Linear Models. A simple linear model. - PowerPoint PPT PresentationTRANSCRIPT
BIOL 582
Lecture Set 6
Tests for multiple groups: Part 2
One-Way ANOVA via the General Linear Model (GLM)
A simple linear model
The ith observation from a sample (i.e., a sample of size n taken from a population of size N (observational study) or randomly assigned (experimental study); this is the observation of one of the subjects in the sample). Y is called the dependent variable.
The y-intercept
Slope
The independent value of the ith subject (e.g., sex, size, rank, category). X is called the independent variable
The portion unexplained by the model (error)
BIOL 582 Intro to Linear Models
A simple linear model
BIOL 582 Intro to Linear Models
0:
0:
1
10
bH
bH
A
This is the model. The model “predicts” or “estimates” a value of y for a given value of x
This is the error or uncertainty of a prediction for any value of y. The model is not perfect. This is a measure of imperfection. For any single value, it is called a residual (error refers more to a treatment of all residuals)
BIOL 582 Intro to Linear Models
A simple linear model
Height (cm)short tall
Mas
s (k
g)li g
hthe
avy
BIOL 582 Intro to Linear Models
A simple linear model
BIOL 582 Linear Models and Hypothesis tests
For a simple linear model,
there are two null hypotheses (but both are rather similar)H0: b0 = 0
H0: b1 = 0
There are a few cases in biological research where testing null hypotheses for intercepts is valuable. We will focus on the slope because the slope has more direct biological meaning (measures a rate of change). Plus, the intercept is rather an artifact of the slope in most cases (a significant negative slope may have a rather large y-intercept). Nevertheless, as the methods for testing both hypotheses are the same, one can easily test the intercept too.
BIOL 582 Linear Models and comparison of group means
The general linear model,
Can also be written as
This is easy to see with a plot
Which indicates that the value of subject j in group i is equal to some expected value (overall mean or group 1 mean) plus some effect, τ, that measures the difference of the ith group from the expected value, plus error. The values of x are 0s and 1s to indicate group association. X is a “dummy variable”
A Group, x B
0 1
Mas
s (k
g)li g
hthe
avy
BIOL 582 Intro to Linear Models
The general linear model for groups
A Group, x B
0 1
Mas
s (k
g)li g
hthe
avy
BIOL 582 Intro to Linear Models
The general linear model for groups
A Group, x B
0 1
Mas
s (k
g)li g
hthe
avy
BIOL 582 Intro to Linear Models
The general linear model for groups
BIOL 582 Linear Models and Hypothesis tests
To avoid unnecessary redundancy, we will try to use this terminology. And to be consistent with the way R uses dummy variables, we will assume that the intercept is the first group mean
Thus the model above indicates that groups differ from the mean of group 1, based on the slope (effect) of b1 for group 2.
The equation, , however, means that τ represents a suite of coefficients for g -1 groups.
This will make more sense soon….
Model + error
BIOL 582 Linear Models and Hypothesis tests
H0: b1 = 0
Testing the null hypothesis for a parameter estimate is the same as comparing two different models! (One model contains the parameter; one lacks it)
“Full” Model
“Reduced” Model (Also called a null model)
Model + error
This is the same as saying a model that only produces the overall mean
This is the same as saying that group means are not different, such that they are equal to each other and equal to the overall mean
BIOL 582 Linear Models and Hypothesis tests
H0: b1 = 0
One-way ANOVA is a comparison of the error variances from full and reduced models!
“Full” Model
“Reduced” Model (Also called a null model)
Is the same as saying
BIOL 582 Linear Models and Hypothesis tests
H0: b1 = 0
One-way ANOVA is a comparison of the error variances from full and reduced models!
Is the same as saying
These are residuals
Therefore
BIOL 582 Linear Models and Hypothesis tests
ˆ( )L β | Y Model likelihood: the likelihood of model parameters (β), given the data (Y)
1 12 2
/ 2
1ˆ( )2
t ti i n
np nL e
Y -Y Σ Y -Y Y μ Σ Y μ
β | YΣ
The right side of the equation is the multivariate normal probability density function (Σ is the error covariance matrix), and is maximized when
(i.e., the exponent is 0). For univariate response data, Σ is simply SSE/nY = μ
1ˆlog ( ) log log 22 2 2
n nL n β Σ This latter part is a constant
WARNING: This may hurt!
The smaller the error, the larger the likelihood of the model
BIOL 582 Linear Models and Hypothesis tests
WARNING: This may hurt!• Consider the following:
•Therefore, is a likelihood ratio test (LRT)stat *
1 1ˆ ˆlog ( ) log ( ) log log2 2full reduced full reducedL L n n
β β Σ Σ
1log log
2 full reducedn Σ Σ
* Note: it is common convention to multiply both sides by -2 and express the LRT as
which approximately follows a Chi-square distribution with (Δk) df, where ki is the number of model parameters for the ith model.
Reduced Model
Reduced Model
Full ModelFull Model
BIOL 582 Linear Models and Hypothesis tests
So What is ANOVA?
ANOVA is a form of likelihood ratio test…..
Consider three ways to test a null hypothesis for a linear model
:
BIOL 582 Comments
It is wise to learn how to perform matrix operations to fully understand linear models
You are not required to do this, but a supplemental slideshow is provided
The assumptions for ANOVA using F are
① Normal error
② Homoscedastic variance
The assumptions for ANOVA using randomization are…. Nothing
The assumptions for the LRT depend on the data used – there are different LRTs, as there are different likelihood functions