| Analysis Of Variance |
Article Index for Analysis Of |
Shopping Variance |
Website Links For Analysis |
Information AboutAnalysis Of Variance |
| CATEGORIES ABOUT ANALYSIS OF VARIANCE | |
| analysis of variance | |
| statistical tests | |
| parametric statisticsanalysis of variance | |
| statistical tests | |
| parametric statistics | |
| statistics | |
|
OVERVIEW There are three conceptual classes of such models:
In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment:
MODELS Fixed-effects model The fixed-effects model of analysis of variance applies to situations in which the experimenter has subjected the experimental material to several treatments, each of which affects only the mean of the underlying normal distribution of the "response variable". Random-effects model Random effects models are used when the treatments are not fixed. This occurs when the various treatments (also known as factor levels) are sampled from a larger population. Because the treatments themselves are Random Variables , some assumptions and the method of contrasting the treatments differ from Anova model 1. Most random-effects or mixed-effects models are not concerned with making inferences concerning the particular sampled factors. For example, consider a large manufacturing plant in which many machines produce the same product. The statistician studying this plant would have very little interest in comparing the three particular machines to each other. Rather, inferences that can be made for ''all'' machines are of interest, such as their variability and the overall mean. ASSUMPTIONS
These together form the common assumption that the error residuals are independently, identically, and normally distributed for fixed effects models, or: : Anova 2 and 3 have more complex assumptions about the expected value and variance of the residuals since the factors themselves may be drawn from a population. LOGIC OF ANOVA Partitioning of the sum of squares The fundamental technique is a partitioning of the total sum of squares into components related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels. : The number of Degrees Of Freedom (abbreviated ) can be partitioned in a similar way and specifies the Chi-square Distribution which describes the associated sums of squares. : The F-test See Also: F-test The F-test is used for comparisons of the components of the total deviation. For example, in one-way, or single-factor Anova, statistical significance is tested for by comparing the F test statistic
:where: :, ''I'' = number of treatments :and :, ''nT'' = total number of cases to the F-distribution with ''I-1'',''nT'' degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the quotient of two mean sums of squares which have a Chi-square Distribution . ANOVA on ranks As first suggested by Conover and Iman in 1981, in many cases when the data do not meet the assumptions of ANOVA, one can replace each original data value by its rank from 1 for the smallest to N for the largest, then run a standard ANOVA calculation on the rank-transformed data. "Where no equivalent nonparametric methods have yet been developed such as for the two-way design, rank transformation results in tests which are more robust to non-normality, and resistant to outliers and non-constant variance, than is ANOVA without the transformation. (Helsel & Hirsch, 2002, Page 177)." However Seaman ''et al.'' (1994) noticed that the rank transformation of Conover and Iman (1981) is not appropriate for testing interactions among effects in a factorial design as it can cause an increase in Type I error (alpha error). Furthermore, if both main factors are significant there is little power to detect interactions
EXAMPLES Group A is given vodka, Group B is given gin, and Group C is given a Placebo . All groups are then tested with a memory task. A one-way ANOVA can be used to assess the effect of the various treatments (that is, the vodka, gin, and placebo). Group A is given vodka and tested on a memory task. The same group is allowed a rest period of five days and then the experiment is repeated with gin. The procedure is repeated using a placebo. A one-way ANOVA with repeated measures can be used to assess the effect of the vodka versus the impact of the placebo. In an experiment testing the effects of expectations, subjects are randomly assigned to four groups: # expect vodka-receive vodka # expect vodka-receive placebo # expect placebo-receive vodka # expect placebo-receive placebo (the last group is used as the Control Group ) Each group is then tested on a memory task. The advantage of this design is that multiple variables can be tested at the same time instead of running two different experiments. Also, the experiment can determine whether one variable affects the other variable (known as Interaction Effects ). A factorial ANOVA (2×2) can be used to assess the effect of expecting vodka or the placebo and the actual reception of either. SEE ALSO
REFERENCES
|
|
|