| 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 |