MEV 019: Unit 12 - Analysis of Variance Tests

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 UNIT 12: ANALYSIS OF VARIANCE TESTS

12.1 Introduction

When comparing two groups, t-tests are appropriate.
But when comparing three or more groups, performing multiple t-tests increases the risk of committing Type I error.
Analysis of Variance (ANOVA) overcomes this problem by testing for differences among several group means simultaneously using variance estimates.

Developed by R.A. Fisher, ANOVA is one of the most widely used statistical tools in scientific research, including agriculture, medicine, social sciences, and environmental studies.

12.2 Objectives

After studying this unit, you will be able to:

  • Explain the concept and purpose of ANOVA.
  • Understand degrees of freedom in ANOVA.
  • State the significance and uses of ANOVA.
  • Perform one-way ANOVA and two-way ANOVA.
  • Recognize the assumptions underlying ANOVA.

12.3 Analysis of Variance (ANOVA)

ANOVA is a statistical method used to test whether the means of k groups are equal.
It partitions the total variability in data into components:

  1. Between-group variability – due to differences in group means.
  2. Within-group variability – due to random error or variation within groups.

The F-test is used to compare these variances.

12.3.1 Significance of Analysis of Variance

  • Determines if observed differences between sample means are statistically significant.
  • Avoids multiple t-tests and controls the overall error rate.
  • Can handle multiple factors simultaneously.

12.3.2 Degrees of Freedom (df)

In ANOVA:

  • Between-groups df = k−1k - 1k−1
  • Within-groups df = N−kN - kN−k
  • Total df = N−1N - 1N−1
    Where:
  • kkk = number of groups
  • NNN = total number of observations

12.3.3 Uses of ANOVA

  • Comparing mean yields of different crop varieties.
  • Assessing effects of different treatments in experiments.
  • Studying variation due to different environmental conditions.
  • Evaluating multiple teaching methods in education research.

12.4 One-way Analysis of Variance (ANOVA)

In one-way ANOVA, there is one factor (independent variable) with two or more levels (groups).
We test whether the means of these groups are equal.

12.4.1 Basic Assumptions of One-Way ANOVA

  1. The populations are normally distributed.
  2. The populations have equal variances (homogeneity of variance).
  3. Samples are independent.

12.4.2 Test of Hypothesis in One-Way ANOVA

Hypotheses:

  • H0H_0H0​: All group means are equal (μ1=μ2=⋯=μk\mu_1 = \mu_2 = \dots = \mu_kμ1​=μ2​=⋯=μk​)
  • H1H_1H1​: At least one mean is different.

Steps:

  1. Calculate group means and overall mean.
  2. Compute Sum of Squares Between (SSB) and Sum of Squares Within (SSW):

SSB=∑i=1kni(Xˉi−Xˉ)2SSB = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X})^2SSB=i=1∑k​ni​(Xˉi​−Xˉ)2 SSW=∑i=1k∑j=1ni(Xij−Xˉi)2SSW = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2SSW=i=1∑k​j=1∑ni​​(Xij​−Xˉi​)2

  1. Find Mean Squares:

MSB=SSBk−1,MSW=SSWN−kMSB = \frac{SSB}{k-1}, \quad MSW = \frac{SSW}{N-k}MSB=k−1SSB​,MSW=N−kSSW​

  1. Compute F-ratio:

F=MSBMSWF = \frac{MSB}{MSW}F=MSWMSB​

  1. Compare F with the critical value from F-distribution table (df₁ = k – 1, df₂ = N – k).
  2. Draw conclusion: Reject H0H_0H0​ if calculated F > critical F.

12.5 Two-way Analysis of Variance (ANOVA)

Two-way ANOVA is used when there are two factors and we want to study:

  • The effect of each factor individually (main effects).
  • The combined effect of both factors (interaction effect).

Example: Studying crop yield influenced by fertilizer type (Factor A) and irrigation level (Factor B).

12.5.1 Basic Assumptions of Two-way ANOVA

  1. Each cell (combination of factors) is normally distributed.
  2. Equal variances across all cells.
  3. Independence of observations.

12.5.2 Test of Hypothesis in Two-way ANOVA

Hypotheses:

  • H0AH_{0A}H0A​: No difference between levels of Factor A.
  • H0BH_{0B}H0B​: No difference between levels of Factor B.
  • H0ABH_{0AB}H0AB​: No interaction between Factor A and B.

Steps:

  1. Arrange data in a two-factor table.
  2. Calculate Sum of Squares for:
    • Factor A (SSA)
    • Factor B (SSB)
    • Interaction (SSAB)
    • Error/Within (SSE)
  3. Find respective Mean Squares by dividing SS by df.
  4. Compute F-ratios for each source:

FA=MSAMSE,FB=MSBMSE,FAB=MSABMSEF_A = \frac{MS_A}{MS_E}, \quad F_B = \frac{MS_B}{MS_E}, \quad F_{AB} = \frac{MS_{AB}}{MS_E}FA​=MSE​MSA​​,FB​=MSE​MSB​​,FAB​=MSE​MSAB​​

  1. Compare with F-critical values for each df set.

12.6 Let Us Sum Up

  • ANOVA compares means of three or more groups using the F-test.
  • One-way ANOVA deals with a single factor; two-way ANOVA handles two factors.
  • Both types require assumptions of normality, equal variances, and independent samples.
  • The F-ratio compares between-group variance to within-group variance to decide significance.

12.7 Key Words

  • ANOVA – Statistical method to compare means of multiple groups.
  • One-way ANOVA – Analysis with one factor.
  • Two-way ANOVA – Analysis with two factors, possibly including interaction effects.
  • Between-group variance – Variation due to differences among group means.
  • Within-group variance – Variation within groups due to random error.
  • F-ratio – Test statistic in ANOVA.

 

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