ASP Statistical Software A STATISTICAL PACKAGE (ASP) For Business, Economics, And The Social Sciences

# Analysis Of Variance

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## Analysis Of Variance Models

ASP allows you to estimate a wide variety of analysis of variance models:

• One-Way Analysis Of Variance models.
• Simple Random Block/Repeated Measures models.
• N Way Analysis Of Variance models with fixed, random, or mixed effects and with or without a nested factor.
• N Way Random Block or Repeated Measures models with fixed, random, or mixed effects and with or without a nested factor.
• N Way Analysis Of Variance/Random Block Covariance models.

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## Basic Analysis Of Variance Output

Output from the ANALYSIS OF VARIANCE routines is generated in two stages:

1. First, an analysis of variance table is displayed.
2. Then you are given the option of generated additional output from the model being estimated:

• Analysis Of Variance Table. The analysis of variance table contains the sum of squares explained by each factor, the error sum of squares, and the total sum of squares along with the degrees of freedom associated with each of these sums of squares.

Next the mean square associated with each sum of squares is listed, as is the F statistic with its associated probability value for the null hypothesis that the factor level means are equal.

Expected mean squares are given for nested designs and for random and mixed models. The denominator of the F statistics in the analysis of variance table is also given for these models, as are the nested sums of squares when appropriate.

The analysis of variance Table for an n-way analysis contains sums of squares, degrees of freedom, mean squares, F statistics, and probability values for the interaction effects between factors as well as for the factors themselves. TOP

## Advanced Analysis Of Variance Output

Once the factor means and effects output is exited you are given options for additional output:

• Means And Effects. This option calculates and displays the factor level effects and means. The estimate of the overall (grand) mean is presented in this display along with the variance of this estimate and the model’s mean squared error and associated degrees of freedom.

Estimates of the effects and means of each factor level are also presented, and the variance of each of these estimates is listed to its right. The number of treatments that were available to estimate each factor level mean are reported.

The output for an n-way analysis reports the effects, means, variances and number for the interactions between individual factor levels as well as for the factor levels themselves.

• Pairwise Comparisons. This option allows you to estimates the values for all possible pairwise comparisons of the factor level and treatment means in the model being estimated. This option allows you to calculate:

• P-Values.
• A Priori Confidence Limits.
• Tukey Confidence Limits.
• Scheffe Confidence Limits.
• Bonferroni Confidence Limits.

The output includes the difference between all possible factor or treatment means, the standard error of each difference, the statistic for the level of confidence you have chosen, this statistic times the standard error, and the lower and upper limits for the confidence interval. A matrix indicating significant differences is also included.

• Linear Combinations. This option allows you to estimate linear combinations of factor or treatment means in the model being estimated. This option allows you to calculate:

• P-Values.
• A Priori Confidence Limits.
• Scheffe Confidence Limits.
• Bonferroni Confidence Limits.

The output for these options includes the value of the linear combination, the standard error of this value, the statistic for the level of confidence you have chosen, this statistic times the standard error, and the lower and upper limits for the confidence interval.

• Listing The Variance/Covariance Matrix. This option allows you to list the variance/covariance matrix of the coefficients of the underlying regression model from which the parameters of the analysis of variance model are derived. The result is an algebraic representation of the underlying analysis of variance regression model along with the variance/covariance matrix for the estimated coefficients in this model.

• List/Plot/Add Residuals To Data Matrix. This option allows you to calculate the standardized, studentized, and jackknife residuals along with the leverage and influence of each residual. These variables are plotted and you are then given the option of adding these variables to the data matrix in the editor. A table of the actual value of the dependent variable, its estimated value, and the residual values is also displayed.

• Breaking Down Residuals. This option allows you to break down and plot the residuals in the model you are estimating by factor and treatment means.

• Plot Treatment Means. This option allows you to break down and plot the treatment means in the model you are estimating by the individual values of the factors.

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## Setting Up ANOVA Regression Matrix

ASP also allows you to setup an analysis of variance regression matrix that can be subsequently altered and estimated by you. This option allows you to specify and estimate a unique general linear model.

When the regression model is estimated the output includes an algebraic representation of the regression equation in this model, and an analysis of variance table that reports the regression, residual, and total sum of squares along with their respective degrees of freedom, the regression and residual mean square, and the F statistic and its probability value for the null hypothesis that the regression sum of squares is zero.

You are also given the option of listing the variance/covariance matrix for this model and of listing, plotting, etc. the residuals of the model.

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## Estimating Factor Means

ASP allows you to estimate the factor means of an analysis of variance model without estimating the model itself. Only the factor and interaction means and variances are estimated by this option and not the factor and interaction effects.

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