QUESTION
Explain ANOVA and its Logic?
- Course: Educational Statistics
- Course code 8614
- Level: B.Ed Solved Assignment
ANSWER
Introduction to Analysis of Variance (ANOVA)
The t-tests have one very serious limitation – they
are restricted to tests of the significance of the difference between only two
groups. There are many times when we like to see if there are significant
differences among three, four, or even more groups. For example, we may want to
investigate which of three teaching methods is best for teaching ninth-class
algebra. In such a case, we cannot use a t-test because more than two groups are involved.
To deal with such types of cases one of the most useful techniques in statistics
is analysis of variance (abbreviated as ANOVA). This technique was developed by British Statistician Ronald A. Fisher (Dietz & Kalof, 2009; Bartz, 1981)
Analysis of Variance (ANOVA) is a hypothesis testing
procedure that is used to evaluate mean differences between two or more
treatments (or populations). Like all other inferential procedures. ANOVA uses
sample data as a basis for drawing general conclusions about populations.
Sometimes, ANOVA and t-tests may be two different ways of doing
exactly the same thing: testing for mean differences. In some cases this is true –
both tests use sample data to test hypotheses about population mean.
However, ANOVA has many more advantages over t-tests. t-tests
are used when we have to compare only two groups or variables (one independent and
one dependent). On the other hand, ANOVA is used when we have two or more two independent variables (treatment). Suppose we want to study the effects of
three different models of teaching on the achievement of students. In this case, we have three different samples to be treated using three different treatments.
So ANOVA is the suitable technique to evaluate the difference.
Logic of ANOVA
Let us take the hypothetical data given in the table.
There are three separate samples, with n = 5 in each sample. The dependent variable is the number of problems solved correctly These data represent results of an independent-measure experiment comparing learning performance under three temperature conditions. The scores are variable and we want to measure the amount of variability (i.e. the size of the difference) to explain where it comes from.
To compare the total variability, we will combine all the scores from all the separate samples into one group and then obtain one general measure of variability for the complete experiment. Once we have measured the total variability, we can begin to break it into separate components. The word analysis means breaking into smaller parts.
Because we are going to analyze the variability, the process is called analysis of variance (ANOVA). This analysis process divides the total variability into two basic components:
i) Between-Treatment Variance
Variance simply means difference and calculating the variance is a process of measuring how big the differences are for a set of numbers. The between-treatment variance measures how much difference exists between the treatment conditions. In addition to measuring differences between treatments, the overall goal of ANOVA is to evaluate the differences between treatments. Specifically, the purpose of the analysis is to distinguish is to distinguish between two alternative explanations.
a) The differences between the treatments have been caused by the treatment effects.
b) The differences between the treatments are simply due to chance.
Thus, there are always two possible explanations for the variance (difference) that exists between treatments
1) Treatment Effect:
The differences are caused by the treatments. the scores in sample 1 are obtained at room temperature of 50o and that of sample 2 at 70o. The difference between samples may be caused by the difference in room temperature.
2) Chance:
The differences are simply due to chance. If there is no treatment effect, even then we can expect some difference between samples. The chance differences are unplanned and unpredictable differences that are not caused or explained by any action of the researcher. Researchers commonly identify two primary sources for chance differences.
Individual Differences
Each participant in the study has their own individual characteristics. Although it is reasonable to expect that different subjects will produce different scores, it is impossible to predict exactly what the difference will be.
Experimental Error
In any measurement, there is a chance of some degree of error. Thus, if a researcher measures the same individuals twice under the same conditions, there is a greater possibility of obtaining two different measurements. Often these differences are unplanned and unpredictable, so they are considered to be by chance.
Thus, when we calculate the between-treatment variance, we are measuring differences that could be either by treatment effect or could simply be due to chance. To demonstrate that the difference is really a treatment effect, we must establish that the differences between treatments are bigger than would be expected by chance alone. To accomplish this goal, we will determine how big the differences are when there is no treatment effect involved. That is, we will measure how much difference (variance) occurred by chance. To measure chance differences, we compute the variance within treatments
ii) Within-Treatment Variance
Within each treatment condition, we have a set of individuals who are treated exactly the same and the researcher does not do anything that would cause these individual participants to have different scores. For example, the data shows that five individuals were treated at a70oroom temperature. Although these five students were all treated exactly the same, their scores are different. The question is why are the scores different? A plain answer is that it is due to chance. the overall analysis of variance and identifies the sources of variability that are measured by each of the two basic components.
Related Topics
What is measure of difference? Explain different types of test
Concept of Reliability, Types and methods of Reliability
Level of Measurement
Types of Variable in Stats
Measures of Central Tedency and Dispersion,
Role of Normal Distribution, and also note on Skewness and Kurtosis.
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