Hypothesis Tests with One-Way ANOVA |
One-way ANOVA models are fit with either an explicit or implicit intercept. In implicit intercept models, the ANOVA table of PROC TRANSREG is the correct table for a model with an intercept, and the regression table is the correct table for a model that does not have a separate explicit intercept. The PROC TRANSREG implicit intercept ANOVA table matches the PROC REG table when the NOINT a-option is not specified, and the PROC TRANSREG implicit intercept regression table matches the PROC REG table when the NOINT a-option is specified. The following statements illustrate this relationship and produce Figure 93.74:
data oneway; input y x $; datalines; 0 a 1 a 2 a 7 b 8 b 9 b 3 c 4 c 5 c ;
title 'Implicit Intercept Model'; proc transreg ss2 data=oneway short; model identity(y) = class(x / zero=none); output out=oneway2; run; proc reg data=oneway2; model y = xa xb xc; /* Implicit Intercept ANOVA */ model y = xa xb xc / noint; /* Implicit Intercept Regression */ run; quit;
Implicit Intercept Model |
Class Level Information | ||
---|---|---|
Class | Levels | Values |
x | 3 | a b c |
Number of Observations Read | 9 |
---|---|
Number of Observations Used | 9 |
Implicit Intercept Model |
Univariate ANOVA Table Based on the Usual Degrees of Freedom | |||||
---|---|---|---|---|---|
Source | DF | Sum of Squares | Mean Square | F Value | Pr > F |
Model | 2 | 74.00000 | 37.00000 | 37.00 | 0.0004 |
Error | 6 | 6.00000 | 1.00000 | ||
Corrected Total | 8 | 80.00000 |
Root MSE | 1.00000 | R-Square | 0.9250 |
---|---|---|---|
Dependent Mean | 4.33333 | Adj R-Sq | 0.9000 |
Coeff Var | 23.07692 |
Univariate Regression Table Based on the Usual Degrees of Freedom | |||||||
---|---|---|---|---|---|---|---|
Variable | DF | Coefficient | Type II Sum of Squares |
Mean Square | F Value | Pr > F | Label |
Class.xa | 1 | 1.00000000 | 3.000 | 3.000 | 3.00 | 0.1340 | x a |
Class.xb | 1 | 8.00000000 | 192.000 | 192.000 | 192.00 | <.0001 | x b |
Class.xc | 1 | 4.00000000 | 48.000 | 48.000 | 48.00 | 0.0004 | x c |
Implicit Intercept Model |
Number of Observations Read | 9 |
---|---|
Number of Observations Used | 9 |
Analysis of Variance | |||||
---|---|---|---|---|---|
Source | DF | Sum of Squares |
Mean Square |
F Value | Pr > F |
Model | 2 | 74.00000 | 37.00000 | 37.00 | 0.0004 |
Error | 6 | 6.00000 | 1.00000 | ||
Corrected Total | 8 | 80.00000 |
Root MSE | 1.00000 | R-Square | 0.9250 |
---|---|---|---|
Dependent Mean | 4.33333 | Adj R-Sq | 0.9000 |
Coeff Var | 23.07692 |
Note: | Model is not full rank. Least-squares solutions for the parameters are not unique. Some statistics will be misleading. A reported DF of 0 or B means that the estimate is biased. |
Note: | The following parameters have been set to 0, since the variables are a linear combination of other variables as shown. |
xc = | Intercept - xa - xb |
---|
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Variable | Label | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| |
Intercept | Intercept | B | 4.00000 | 0.57735 | 6.93 | 0.0004 |
xa | x a | B | -3.00000 | 0.81650 | -3.67 | 0.0104 |
xb | x b | B | 4.00000 | 0.81650 | 4.90 | 0.0027 |
xc | x c | 0 | 0 | . | . | . |
Implicit Intercept Model |
Number of Observations Read | 9 |
---|---|
Number of Observations Used | 9 |
Note: | No intercept in model. R-Square is redefined. |
Analysis of Variance | |||||
---|---|---|---|---|---|
Source | DF | Sum of Squares |
Mean Square |
F Value | Pr > F |
Model | 3 | 243.00000 | 81.00000 | 81.00 | <.0001 |
Error | 6 | 6.00000 | 1.00000 | ||
Uncorrected Total | 9 | 249.00000 |
Root MSE | 1.00000 | R-Square | 0.9759 |
---|---|---|---|
Dependent Mean | 4.33333 | Adj R-Sq | 0.9639 |
Coeff Var | 23.07692 |
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Variable | Label | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| |
xa | x a | 1 | 1.00000 | 0.57735 | 1.73 | 0.1340 |
xb | x b | 1 | 8.00000 | 0.57735 | 13.86 | <.0001 |
xc | x c | 1 | 4.00000 | 0.57735 | 6.93 | 0.0004 |