IGNOU solved assignments december

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Here I am giving you IGNOU MS-95 Solved Assignment December :

1.Explain the meaning of "Analysis of Variance". Describe briefly the techniques of analysis of variance for two-way classifications.

Analysis of Variance . A statistical technique which helps in making inference whether three or more samples might come from populations having the same mean; specifically, whether the differences among the samples might be caused by chance variation It is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables.
EXAMPLE -take a sample of 5000 people, who never drank red wine for 10 years. -take another sample of 5000 people who drank red wine 2 glasses everyday over the 10 years.

Check the mean life span of the two groups. The second group mean life span was 78 years, as against the first group, whose mean life span was only 64 years.

Analysis of Covariance (ANCOVA):

Analysis of covariance is a more sophisticated method of analysis of variance. It is based on inclusion of supplementary variables (covariates) into the model. This lets you account for inter-group variation associated not with the "treatment" itself, but with covariate(s). Suppose you analyze the results of a clinical trial of three types of treatment of a disease - "Placebo", "Drug 1", and "Drug 2". The results are three sets of survival times, corresponding to patients from the three treatment groups. The question of interest is whether there is a difference between the three types of treatment in the average survival time. You might use analysis of variance to answer this question. But, if you have supplementary information, for example, each patient's age, then analysis of covariance allows you to adjust the treatment effect (survival time, in this case) to a particular age, say, the mean age of all patients. Age in this case is a "covariate" - it is not related to treatment, but can affect the survival time. This adjustment allows you to reduce the observed variation between the three groups caused not by the treatment itself but by variation of age. If the covariate(s) are associated with the treatment effect, then analysis of covariance may have more power than analysis of variance.

COVARIANCE

The main purpose af the analysis of covariance is statistical control of variability when experimental control can not be used. It is a statistical method for reducing experimental error or for removing the effect of an extraneous variable. Statistical control is obtained by using a concomitant variable (called the covariate) along with the dependent variable. The analysis of covariance makes use of linear prediction or regression as described in chapter one under "Linear Prediction." The underlying rational for the analysis of covariance is the idea of using prediction equations to predict the values (scores and means) of the dependent variable on the basis of the values of the covariate variable, and then subtracting these predicted scores and means from the corresponding values of the dependent variable. Oversimplifying a bit, the analysis of covariance is like an analysis of variance on the residuals of the values of the dependent variable, after removing the influence of the covariate, rather than on the original values
themselves. In so far as the measures of the covariate are taken in advance of the experiment and they correlate with the measures of the dependent variable they can be used to reduce experimental error (the size of the error term) or control for an extraneous variable by removing the effects of the covariate from the dependent variable. The two-way analysis of variance is an extension to the one-way analysis of variance. There are two independent variables (hence the name two-way).

Assumptions
• The populations from which the samples were obtained must be normally or approximately normally distributed.
• The samples must be independent.
• The variances of the populations must be equal.
• The groups must have the same sample size.

Hypotheses
There are three sets of hypothesis with the two-way ANOVA. The null hypotheses for each of the sets are given below. The population means of the first factor are equal. This is like the one-way ANOVA for the row factor. The population means of the second factor are equal. This is like the one-way ANOVA for the column factor. There is no interaction between the two factors. This is similar to performing a test for independence with contingency tables.

Factors
The two independent variables in a two-way ANOVA are called factors. The idea is that there are two variables, factors, which affect the dependent variable. Each factor will have two or more levels within it, and the degrees of freedom for each factor is one less than the number of levels.

Treatment Groups
Treatement Groups are formed by making all possible combinations of the two factors. For example, if the first factor has 3 levels and the second factor has 2 levels, then there will be 3x2=6 different treatment groups. As an example, let's assume we're producing a pharma product. The type of chem. A and type of chem. B are the two factors we're considering in this example. This example has 15 treatment groups. There are 3-1=2 degrees of freedom for the type of CHEM A, and 5-1=4 degrees of freedom for the type of CHEM B. There are 2*4 = 8 degrees of freedom for the interaction between the type of CHEM A and type of CHEM B.

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