CR-p -- Fixed Effects Model
Schematic with Example Data
Level a1
a2 a3 a4 Total
4
6
3
3
1
3
2
2
4
5
4
3
2
3
4
3
5
6
5
4
3
4
3
4
3
5
6
5
6
7
8
10
Mean 3.0
3.5 4.25 6.25 4.25
sd 1.51 0.93 1.04 2.12 1.88
Or in abbreviated form:
Level a1
a2 a3 a4 Total
S1
n=8
S2
n=8
S3
n=8
S4
n=8
Mean 3.0
3.5 4.25 6.25 4.25
sd 1.51 0.93 1.04 2.12 1.88
Where each Sj is an independent randomly assigned group of subjects.
Linear Model
Yij is the score for the ith observation in the
jth treatment level
Y'j is the predicted value for the jth treatment level and is
equal to the mean of the group
μ is the overall population mean (grand mean)
αj is the effect of A treatment level j which is equal to
μj - μ and is subject to the
restriction that Σαj = 0 over j
εi(j) is the error effect associated with Yij
and is equal to Yij - μ - αj .
The error effect is a random variable that is
distributed NID(0,s2ε)
Hypotheses
1. The linear model reflects all sources of variation.
2. The experiment contains all the treatment levels of interest.
3. The εi(j) are independent of each other.
4. The εi(j) are normally distributed.
5. The εi(j) have equal variance in the population.
Notes:
Assumptions 1 & 2 are concerned with model specification.
Because μ and αj are constants, the following holds
ANOVA Summary Table
Source | SS | df | MS | F | p-value | |
Between Groups | 49.0 | 3 | 16.333 | 7.50 | 0.0008 | |
Within Groups | 61.0 | 28 | 2.179 | |||
Total | 110.0 | 31 |
The ANOVA Summary Table may also look like this:
Source | SS | df | MS | F | p-value | |
Treatment | 49.0 | 3 | 16.333 | 7.50 | 0.0008 | |
Error | 61.0 | 28 | 2.179 | |||
Total | 110.0 | 31 |
Expected Mean Squares
Correctly Formed F-ratios
Table of Group Means and Variances
a1 | a2 | a3 | a4 | |
Mean | 3.00 | 3.50 | 4.25 | 6.25 |
Variance | 2.29 | 0.86 | 1.07 | 4.50 |
Std Dev | 1.51 | 0.93 | 1.04 | 2.12 |
A Measure of Strength of Association
Omega-squared (ω2) is the recommended measure of strength of association for fixed-effects analysis of variance models.
From the Example:
49 - (3)2.179 ω2 = --------------- = 0.3785 110 + 2.179
The following guidelines are suggested by Cohen (1989):
In terms of the fhat index of effect size:
Note: The fhat index of effect size should not be confused with Cohen's d index of effect size. The fhat index is derived directly form the ω2.
Model for Orthogonal Coding
G X1 X2 X3 1 1 1 1 2 -1 1 1 3 0 -2 1 4 0 0 -3
Stata Computer Example
input y grp x1 x2 x3 4 1 1 1 1 6 1 1 1 1 3 1 1 1 1 3 1 1 1 1 1 1 1 1 1 3 1 1 1 1 2 1 1 1 1 2 1 1 1 1 4 2 -1 1 1 5 2 -1 1 1 4 2 -1 1 1 3 2 -1 1 1 2 2 -1 1 1 3 2 -1 1 1 4 2 -1 1 1 3 2 -1 1 1 5 3 0 -2 1 6 3 0 -2 1 5 3 0 -2 1 4 3 0 -2 1 3 3 0 -2 1 4 3 0 -2 1 3 3 0 -2 1 4 3 0 -2 1 3 4 0 0 -3 5 4 0 0 -3 6 4 0 0 -3 5 4 0 0 -3 6 4 0 0 -3 7 4 0 0 -3 8 4 0 0 -3 10 4 0 0 -3 end tabstat y, by(grp) stat(n mean sd var) Summary for variables: y by categories of: grp grp | N mean sd variance ---------+---------------------------------------- 1 | 8 3 1.511858 2.285714 2 | 8 3.5 .9258201 .8571429 3 | 8 4.25 1.035098 1.071429 4 | 8 6.25 2.12132 4.5 ---------+---------------------------------------- Total | 32 4.25 1.883716 3.548387 -------------------------------------------------- display 2.12132/.9258201 2.2912875 histogram y, by(grp) normal
robvar y, by(grp) /* W0 is Levene's test of homoscedasticity */ | Summary of y grp | Mean Std. Dev. Freq. ------------+------------------------------------ 1 | 3 1.5118579 8 2 | 3.5 .9258201 8 3 | 4.25 1.0350983 8 4 | 6.25 2.1213203 8 ------------+------------------------------------ Total | 4.25 1.8837163 32 W0 = 1.292876 df(3, 28) Pr > F = .29625408 W50 = 1.037037 df(3, 28) Pr > F = .39138742 W10 = 1.292876 df(3, 28) Pr > F = .29625408 anova y grp Number of obs = 32 R-squared = 0.4455 Root MSE = 1.476 Adj R-squared = 0.3860 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 49.00 3 16.3333333 7.50 0.0008 | grp | 49.00 3 16.3333333 7.50 0.0008 | Residual | 61.00 28 2.17857143 -----------+---------------------------------------------------- Total | 110.00 31 3.5483871 /* user written program -- findit effectsize */ effectsize grp anova effect size for grp with dep var = y total variance accounted for omega2 = .37854187 eta2 = .44545455 Cohen's f = .78046067 partial variance accounted for partial omega2 = .37854187 partial eta2 = .44545455 /* Tukey-Kramer pairwise comparisons */ /* user written program -- findit tkcomp */ tkcomp grp Tukey-Kramer pairwise comparisons for variable grp studentized range critical value(.05, 4, 28) = 3.8613586 mean grp vs grp group means dif TK-test ------------------------------------------------------- 1 vs 2 3.0000 3.5000 0.5000 0.9581 1 vs 3 3.0000 4.2500 1.2500 2.3954 1 vs 4 3.0000 6.2500 3.2500 6.2279* 2 vs 3 3.5000 4.2500 0.7500 1.4372 2 vs 4 3.5000 6.2500 2.7500 5.2698* 3 vs 4 4.2500 6.2500 2.0000 3.8326 oneway y grp, noanova sidak bonferroni scheffe Comparison of y by grp (Bonferroni) Row Mean-| Col Mean | 1 2 3 ---------+--------------------------------- 2 | .5 | 1.000 | 3 | 1.25 .75 | 0.608 1.000 | 4 | 3.25 2.75 2 | 0.001 0.005 0.068 Comparison of y by grp (Scheffe) Row Mean-| Col Mean | 1 2 3 ---------+--------------------------------- 2 | .5 | 0.927 | 3 | 1.25 .75 | 0.427 0.794 | 4 | 3.25 2.75 2 | 0.002 0.009 0.085 Comparison of y by grp (Sidak) Row Mean-| Col Mean | 1 2 3 ---------+--------------------------------- 2 | .5 | 0.985 | 3 | 1.25 .75 | 0.474 0.900 | 4 | 3.25 2.75 2 | 0.001 0.005 0.066 /* regression with orthogonal coding */ regress y x1 x2 x3 Source | SS df MS Number of obs = 32 -------------+------------------------------ F( 3, 28) = 7.50 Model | 49.00 3 16.3333333 Prob > F = 0.0008 Residual | 61.00 28 2.17857143 R-squared = 0.4455 -------------+------------------------------ Adj R-squared = 0.3860 Total | 110.00 31 3.5483871 Root MSE = 1.476 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x1 | -.25 .3689996 -0.68 0.504 -1.005861 .5058614 x2 | -.3333333 .213042 -1.56 0.129 -.7697301 .1030635 x3 | -.6666667 .1506435 -4.43 0.000 -.9752458 -.3580875 _cons | 4.25 .2609221 16.29 0.000 3.715525 4.784475 ------------------------------------------------------------------------------ /* regression with dummy coding */ regress y i.grp Source | SS df MS Number of obs = 32 -------------+------------------------------ F( 3, 28) = 7.50 Model | 49 3 16.3333333 Prob > F = 0.0008 Residual | 61 28 2.17857143 R-squared = 0.4455 -------------+------------------------------ Adj R-squared = 0.3860 Total | 110 31 3.5483871 Root MSE = 1.476 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- grp | 2 | .5 .7379992 0.68 0.504 -1.011723 2.011723 3 | 1.25 .7379992 1.69 0.101 -.2617229 2.761723 4 | 3.25 .7379992 4.40 0.000 1.738277 4.761723 | _cons | 3 .5218443 5.75 0.000 1.93105 4.06895 ------------------------------------------------------------------------------ /* cell means using margins command */ margins grp Adjusted predictions Number of obs = 32 Model VCE : OLS Expression : Linear prediction, predict() ------------------------------------------------------------------------------ | Delta-method | Margin Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- grp | 1 | 3 .5218443 5.75 0.000 1.977204 4.022796 2 | 3.5 .5218443 6.71 0.000 2.477204 4.522796 3 | 4.25 .5218443 8.14 0.000 3.227204 5.272796 4 | 6.25 .5218443 11.98 0.000 5.227204 7.272796 ------------------------------------------------------------------------------
Some Formulas
Recall the linear model,
The grand mean is the general level of scores,
The treatment effect is the elevation or depression of scores due to the jth treatment,
The error effect is unique to subject i in treatment level j,
The above implies,
From the prediction model (way above),
Partitioning Sums of Squares
Linear Statistical Models Course
Phil Ender, 17sep10, 11apr06, 12Feb98