Variance analysis

(noun)

The difference between a budgeted, planned, or standard amount and the actual amount incurred/sold. This difference can be computed for both costs and revenues.

Examples of Variance analysis in the following topics:

  • Exercises

    • What is the null hypothesis tested by analysis of variance?
    • Why not just compute t-tests among all pairs of means instead computing an analysis of variance?
    • How is it that estimates of variance can be used to test a hypothesis about means?
    • Give the source and degrees of freedom columns of the analysis of variance summary table.
    • Compute the analysis of variance summary table.

  • Graphical diagnostics for an ANOVA analysis

    • There are three conditions we must check for an ANOVA analysis: all observations must be independent, the data in each group must be nearly normal, and the variance within each group must be approximately equal.
    • The last assumption is that the variance in the groups is about equal from one group to the next.
    • Independence is always important to an ANOVA analysis.
    • The constant variance condition is especially important when the sample sizes differ between groups.

  • Introduction

    • Discuss two uses for the F distribution: One-Way ANOVA and the test of two variances.
    • For hypothesis tests involving more than two averages, statisticians have developed a method called Analysis of Variance"(abbreviated ANOVA).
    • You will also study the F distribution, used for One-Way ANOVA, and the test of two variances.

  • ANOVA Assumptions

    • This design-based analysis was developed by Francis J.
    • Therefore, by contraposition, a necessary condition for unit-treatment additivity is that the variance is constant.
    • In summary, the normal model based ANOVA analysis assumes the independence, normality and homogeneity of the variances of the residuals.
    • The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment.
    • Both these analyses require homoscedasticity, as an assumption for the normal model analysis and as a consequence of randomization and additivity for the randomization-based analysis.

  • Two-mode factor analysis

    • Factor analysis provides an alternative method to SVD to the same goals: identifying underlying dimensions of the joint space of actor-by-event variance, and locating or scaling actors and events in that space.
    • The method used by factor analysis to identify the dimensions differs from SVD.
    • Figure 17.10 shows the eigenvalues (by principle components) calculated by Tools>2-Mode Scaling>Factor Analysis.
    • This solution, although different from SVD, also suggests considerable dimensional complexity in the joint variance of actors and events.
    • The factor analysis method does produce somewhat lower complexity than SVD.

  • Proportion of Variance Explained

    • In this example, the variance of scores is 2.794.
    • We estimate this by computing the variance within each of the treatment conditions and taking the mean of these variances.
    • For this example, the mean of the variances is 2.649.
    • Since with Design 1 the variance due to Dose would be smaller and the total variance would be larger, the proportion of variance explained by Dose would be much less using Design 1 than using Design 2.
    • The sources of variation, degrees of freedom, and sums of squares from the analysis of variance summary table as well as four measures of effect size are shown in Table 3.

  • Variance and standard deviation

    • Both of these are very useful in data analysis, even though their formulas are a bit tedious to calculate by hand.
    • The standard deviation is defined as the square root of the variance:
    • The variance is roughly the average squared distance from the mean.
    • The standard deviation is the square root of the variance.
    • The σ2 population variance and for the standard deviation.

  • Two-mode correspondence analysis

    • For binary data, the use of factor analysis and SVD is not recommended.
    • Factoring methods operate on the variance/covariance or correlation matrices among actors and events.
    • Correspondence analysis (rather like Latent Class Analysis) operates on multi-variate binary cross-tabulations, and its distributional assumptions are better suited to binary data.
    • We do see, however, that this method also can be used to locate the initiatives along multiple underlying dimensions that capture variance in both actors and events.
    • Figure 17.15 show the plot of the actors and events in the first two dimensions of the joint correspondence analysis space.

  • Regression Analysis for Forecast Improvement

    • Regression Analysis is a causal / econometric forecasting method that is widely used for prediction and forecasting improvement.
    • Regression Analysis is a causal / econometric forecasting method.
    • The errors are uncorrelated, that is, the variance– co-variance matrix of the errors is diagonal, and each non-zero element is the variance of the error.
    • The variance of the error is constant across observations (homoscedasticity).
    • A large body of techniques for carrying out regression analysis has been developed.

  • Optimization by Tabu search

    • Tabu search does this by searching for sets of actors who, if placed into a blocks, produce the smallest sum of within-block variances in the tie profiles.
    • That is, if actors in a block have similar ties, their variance around the block mean profile will be small.
    • So, the partitioning that minimizes the sum of within block variances is minimizing the overall variance in tie profiles.
    • This last analysis illustrates most fully the primary goals of an analysis of structural equivalence: