Analysis of Variance (ANOVA)-II. Subscribe my youtube channel "Biology with kiran"

Analysis of Variance (ANOVA) is a statistical technique used to determine if there are any statistically significant differences between the means of three or more independent (unrelated) groups. The basic idea is to test for significant differences in means by comparing variances.

Key Concepts in ANOVA
Null Hypothesis (
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0
H
0
​
): Assumes that there are no differences between the group means. In other words, any observed differences are due to random chance.

Alternative Hypothesis (
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H
a
​
): Assumes that at least one group mean is different from the others.

Between-Group Variance: Measures the variation between the different group means. If the group means are significantly different, this variance will be large.

Within-Group Variance: Measures the variation within each group. This is the inherent variability of the data within each group.

F-Statistic: The ratio of the between-group variance to the within-group variance. A higher F-statistic indicates that the between-group variability is much larger than the within-group variability, suggesting significant differences among group means.

Steps in Conducting ANOVA
Calculate the Group Means: Compute the mean of each group.

Compute the Overall Mean: Calculate the mean of all data points combined.

Sum of Squares Between (SSB): Measure the total deviation of each group mean from the overall mean, weighted by the number of observations in each group.

Sum of Squares Within (SSW): Measure the total deviation of each observation from its respective group mean.

Total Sum of Squares (SST): Sum of SSB and SSW. This represents the total variation in the data.

Degrees of Freedom:

Between-group degrees of freedom (
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df
between
​
): Number of groups minus one.
Within-group degrees of freedom (
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ℎ
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df
within
​
): Total number of observations minus the number of groups.
Mean Squares:

Mean Square Between (MSB) = SSB /
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Mean Square Within (MSW) = SSW /
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​

Calculate the F-Statistic:
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=
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F=
MSW
MSB
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Compare the F-Statistic to the Critical Value: Determine if the F-statistic is significant by comparing it to a critical value from the F-distribution table based on the degrees of freedom and chosen significance level (
𝛼
α).

P-Value: Alternatively, calculate the p-value associated with the F-statistic. If the p-value is less than the chosen significance level, reject the null hypothesis.

1. TWO -WAY ANOVA how to perform in EXCEL Two way ANOVA Table Source  SS df Mean Square F Factor  A   SS(A) (a−1) SS(A)/(a−1) Mean Square (A) / Mean Square Error Factor  B   SS(B) (b−1) SS(B)/(b−1) Mean Square (B) / Mean Square Error Interaction  (AB) SS(AB) (a−1)(b−1) SS(AB)/((a−1)(b−1)) Mean Square (AB) / Mean Square Error Error  SSE (N−ab) SSE/(N−ab)   Total (Corrected)  SS(Total) (N−1)    

2. Analysis of Variance (ANOVA) Analysis of variance (ANOVA) - separates observed variance data into different components TWO -WAY ANOVA A two-way ANOVA tests the effect of two independent variables on a dependent variable. Examples: 1. Effect of fertilizer (factor A) and pesticide (factor B) application Grain yield (dependent variable) of wheat 2. Effect of feed (factor A) and rearing temperature (factor B) on weight gain ( dependent variable ) of insects ANOVA Table Source  SS df Mean Square F Factor  A   SS(A) (a−1) SS(A)/(a−1) Mean Square (A) / Mean Square Error Factor  B   SS(B) (b−1) SS(B)/(b−1) Mean Square (B) / Mean Square Error Interaction  (AB) SS(AB) (a−1)(b−1) SS(AB)/((a−1)(b−1)) Mean Square (AB)/ Mean Square Error Error  SSE (N−ab) SSE/(N−ab)   Total (Corrected)  SS(Total) (N−1)     An experiment that utilizes every combination of factor levels as treatments is called a factorial experiment.

3. Analysis of Variance (ANOVA) Two Way ANOVA table- Interpretation The test statistic:  F  value for Factor A = 65.40 (α of 0.05, we have F 0.05;2,12  = 4.75 ) F value for Factor B = 5.70 (α of 0.05, we have F 0.05;2,12  = 3.88 ) F value for Factor AB = 0.46 (α of 0.05, we have F 0.05;2,12  = 3.88 ) If test statistic is > the critical value then We reject the null hypothesis (that population means are equal) Conclusion: For factor A and Factor B- Significant difference in means exists Source of Variation SS df MS F P-value F crit Sample 9.7534 1 9.7534 65.3937 0.000000003 4.747225 Columns 1.7003 2 0.85015 5.7000 0.018188275 3.885294 Interaction 0.1376 2 0.06882 0.461 0.641114808 3.885294 Within 1.7898 12 0.14915 Total 13.3812 17         ANOVA table results

4. Analysis of Variance (ANOVA) Partition response into components The goal in this procedure is to split the total variation in the data into a portion due to random error and portions due to changes in the values of the independent variable(s). A treatment is a specific combination of factor levels whose effect is to be compared with other treatments . ANOVA Table Sources of Variation SS DF MS F Treatments (between) SST k−1 SST/(k−1) MST/MSE Error (within) SSE N−k SSE/(N−k)   Total (corrected) SS N−1     The word "source" stands for source of variation. Some authors prefer to use "between" and "within" instead of "treatments" and "error", respectively. If no true variance exists between the groups, the ANOVA's F-ratio should equal close to 1 .