When one designates analysis of variance (actually analyses of variance) a large group of dataanalytic and sample-recognizing statistic procedures, which permit numerous different applications. It is common to them that they analyze the variance, in order to attain explanations about the regularities fitted behind the data. The analysis of variance is called in many computer programs also ANOVA for analysis OF variance.

Not all procedures, which count on the variance, are called analysis of variance, but only such, which explains the variance of a metric variate by the influence of one or several group variables. A test statistic is always computed. Regarding the variables involved essentially four forms of the analysis of variance can be differentiated:

• in-factorial univariate analysis of variance
• more-factorial univariate analyses of variance
• in-factorial multivariate analyses of variance
• more-factorial multivariate analyses of variance

Examples of the application of the analysis of variance are the investigation of the effectiveness of medicines in the medicine (see double-blind attempt) and the investigation of the influence of fertilizers on the yield of cultivated areas in the agriculture.

Philosophy of the analyses of variance

The procedures examine whether (and if necessary as) the expectancy values of the metric variates differ in different groups (also classes). With the test statistics of the procedure it is tested whether the variance between the groups is larger than the variance within the groups. Thus it can be determined whether the group organization is meaningful or not and/or whether the groups differ significantly or not.

If they differ significantly, it can be accepted that in the groups different regularities work. Thus it can be clarified for example whether the "“behavior"” of a control's group is identical to that an experimental's group. For example if the variance of one of these two groups is led on material causes (sources of variance) back already, it can be closed with variance equality that in the other group no new effect cause (e.g. by the experimental conditions) came in addition.

See also: Diskriminanzanalyse, null hypothesis, certainty measure

Meaning of the analysis of variance

The analysis of variance plays a substantial role in the science. It can be regarded as the scientifically founded form of the Attribuierung (cause writing up), which humans in naive way in the everyday life to constantly operate and frequentnesses and variability of actions or procedures on presumed reasons back to lead. Everyday life procedures, which arise often together or varieren/kovariieren under environment changes in similar way, are interpreted standing in the connection as with one another. One accepts variations in the everyday life, decreased/went back on dependence. So the frequency (intensity, way etc.) can, with which a person their hands washes, with the frequency in purchase to be set, with which it implements dirty activities. (Garage mechanics) in the everyday life naively the conclusion is then drawn that a person, who washes straight its hands had before a contamination, which is regarded as reason for the hand washing. This Usachenzuschreibung is however only as a rule correct, since there are also different reasons for hand washing. Everyday life observations can be deceitful.

Therefore a substantial role comes to the statistic analysis of variance, since it continues the everyday life thinking in consistent form. Many other multivariate procedures continue the everyday life thinking not, but are based in artificially developed model acceptance.

The significance of a determined group organization can be tested on the basis the F-Verteilung. The values in this distribution are the test statistic of the analysis of variance.

Conditions of the analyses of variance

The application of each form of the analysis of variance is bound at conditions, whose being present before each computation must be examined. If the data records do not fulfill these conditions, then the results are useless. The conditions are the following depending upon application somewhat differently, generally apply:

• Variance homogeneity of the sample variables
• Normal distribution of the sample variables

The examination takes place with other test outside of the analysis of variance, which are provided however today according to standard in statistical programs as option. The normal distribution can be examined for example for each variable with the Kolmogorow Smirnow test. If these conditions are not fulfilled, offer themselves distribution free ones, non parametric procedure, which is durable, but count less exactly.

Terms

• Dependent variable aV (goal variable)
  • The metric variable, whose value is to be explained by the kategorialen variables. This variable contains measured values.
• Argument UV (influence variable, factors)
  • The kategoriale variable, which gives the groups. Their influence is to be examined.
  • Also factor mentioned. The categories are called then factor stages. This designation is not identical to that one with the factor analysis.

A-factorial analysis of variance

With a in-factorial analysis of variance one examines the influence argument (factor) with p different developments for a dependent variable, which contains the measured values.

Conditions

• The error components must be normaldistributed. Error components designate the respective variances (entire, Treatment and error variance). The validity of this condition presupposes at the same time a normal distribution of the measured values in the respective population.
• The error variances must be between the groups (thus the factor stages) same and/or homogeneous
• The measured values and/or factor stages must be independent.

Example

This form of the analysis of variance is indicated, if it is to be examined whether smoking has an influence on the aggressiveness. Smokes is here an argument, which can be divided into three developments (factor stages): Nonsmoker, weak smokers and strong smokers. The aggressiveness seized by a questionnaire is the dependent variable. For the execution of the investigation the Vpn is assigned to the three groups. Afterwards the questionnaire is submitted, with which the aggressiveness is seized.

Hypotheses

The null hypothesis of a in-factorial analysis of variance reads:

H_0: \ mu_1 = \ mu_2 ="… = \ mu_p

The alternative hypothesis reads:

H_1: \ mu_i \ neq \ mu_j

Null hypothesis therefore means that between the average values of the groups (those factor development and/or factor stages correspond) no difference exists. The alternative hypothesis means that between at least two average values a difference exists. If we have for example five factor stages, the alternative hypothesis is confirmed if at least two of the group average values differ. Can differ in addition, three average values or four or all five clearly from each other.

If null hypothesis is rejected, the analysis of variance supplies with thus neither to explanation about it, between like many still between which factor stages a difference exists. We know then only with a certain probability (see level of significance, error 1. Kind) that at least two developments exhibit an important difference.

One can ask now whether it would be permissible to accomplish with different t-tests in pairs in each case single comparisons between the average values. If one compares only two groups (thus two average values) with the analysis of variance, then lead t-test and analysis of variance to the same result. Couches however more than two groups forwards, the examination of the global Nullypothese of the analysis of variance is illegal over t-tests in pairs - it comes to the so-called alpha error cumulating.

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