Analysis of Variance (ANOVA) – Repeated Measures

What is ANOVA repeated-Measure?

Repeated Measures

1. An independent variable is manipulated to create two or more treatment conditions, with the same group of participants compared in all of the experiments.
2. Study with the same group of individuals by observing at two or more different times.

ANOVA repeated-Measures: Assumptions 

1. The observations within each treatment condition must be independent
2. The population distribution within each treatment must be normal.
3. The variances of the population distributions for each treatment should be equivalent

ANOVA repeated-Measures: Hypothesis test

Null hypothesis: ANOVA repeated-Measures

• There are no mean differences among the treatment conditions being compared.

H₀ : μ1 = μ2 = μ3 = …………………..

Alternative hypothesis: ANOVA repeated-Measures

• There are mean differences among the treatment conditions.
• At least one treatment mean (μ) is different from another.

H1 : At least one treatment mean (μ) is different

ANOVA repeated-Measures: F-Ratio

• A large value for the F-ratio calculated indicates that the differences between treatments are greater.

• If the F-ratio calculated is larger than the critical value in the F distribution table, differences between treatments are significantly larger.
• In a repeated-measures ANOVA, the denominator of the F-ratio is called the residual variance, or the error variance, and measures how much variance is expected if there are no systematic treatment effects and no individual differences contributing to the variability of the output.
• The first stage of the repeated-measures ANOVA is identical to the independent-measures analysis and separates the total variability into two components: between-treatments and within-treatments.
• Because a repeated-measures design uses the same subjects in every treatment condition, the differences between treatments cannot be caused by individual differences. Thus, individual differences are automatically eliminated from the between-treatments variance in the numerator of the F-ratio.

ANOVA: Repeated-Measures: Effect size

Example: 6 students selected from the same section of the class. As per students’ past records, they are good in academics. They have joined Physics, Chemistry & Maths tuitions with three different coaching institute’s teacher subject wise. After successful completion of the course, they appeared in the exam and their scores are recorded in a table (refer to below table)

Solution:

1. Calculate Physics, Chemistry & Maths total score ( T = 40 , 66 , 74 )

• Calculate Physics, Chemistry & Maths Grand total score G :

G = T1 + T2 + T3

G = 40 + 66 + 74

G = 180

• ---

  Calculate every student’s total score (Physics + Chemistry + Maths) P :

P1=21 , P2=24 , P3=36 , P4=36 , P5=30 , P6=33

• ---

  Calculate Subject scores mean M :

• ---

  Calculate Subjects SS :

SS_physics = 45.3 , SS_chemistry = 34.0 , SS _maths = 23.3

• ---

  Calculate Subjects X² :

X² =  sum (X * X) = 2008

• ---

  ANOVA Summary :

ANOVA: Repeated-Measures: Analysis :

• The total sum of square: SStotal

• df total

• The sum of the square within treatment: SSwithin

• Degree of freedom within treatment: df within

• The sum of square between SSbetween

• Degree of freedom between teachers(/subjects)

• The sum of squares between students: df between students

• degree of freedom between students: df between students

• Error:

SS_error = SS_{within teacher (subjects)} – SS_{between students}

SS_error = 102.7 – 66

SS_error = 36.7

• Degree of freedom of error: df error

df_error = df_{within treatments} – df_{between students}

df_error = 15 – 5

df_error = 10

ANOVA repeated-Measures: Mean square value (MS)

• MS-between Teachers (subjects)

• MS-Error

• F-ratio calculation 

• F-ratio from table

  • Select the F value from the table  with df of the denominator and  df of the numerator

At α = 0.05 level,

F (10, 2) = 4.10

• • Calculated  F = 14.35, is well beyond the critical values at α = .05

• • • 
  • Can be concluded that H₀ is rejected
  • Differences between teachers (subjects) are significantly greater than expected by chance.

ANOVA Repeated-Measures: Analysis summary

ANOVA repeated-Measures: Measuring Effect Size

• The percentage of variance accounted for by the treatment effect is usually called η² (the Greek letter eta squared) instead of using r²

The percentage of variance accounted for =η² 

• MS-between Teachers (subjects) Calculation

• η^{2 =} 95% means variability in the scores is because of differences between treatments – Teacher (subjects).

ANOVA: Repeated-Measures: Post hoc tests

• If ANOVA analysis rejects the null hypothesis. Then additional analyses are conducted to determine exactly which treatments are significantly different and which are not.
• These analyses are called post hoc tests.
• Post-analysis tests are Tukey’s HSD test and the Scheffé test.

ANOVA: Repeated measures: Tukey’s Honestly Significant Difference (HSD) Test

• This test computes a single value that determines the minimum difference between treatment means that is necessary for significance.
• This value is called the honestly significant difference, or HSD
• It is used to compare any two treatment conditions.
• If the mean difference exceeds Tukey’s HSD, it is concluded that there is a significant difference between the treatments.

Treatment effect: Tukey’s HSD

ANOVA: Repeated-Measures: The Scheffè Test

• This test uses an F-ratio to evaluate the significance of the difference between any two treatment conditions.
• The numerator of the F-ratio is an MS between treatments that are calculated using only the two treatments under comparison.
• The denominator is the same MS_within that was used for the overall ANOVA.
• If F is calculated > F table, then there is a significant difference between these two treatments.

ANOVA repeated-Measures: Benefits

• In ANOVA-independent measures, the presence of a treatment effect is masked by the influence of individual differences.
• This problem is eliminated by ANOVA-repeated measures. In this method, individual differences is separated out.
• When the individual differences are large, ANOVA repeated-measures analysis provides a more sensitive test for a treatment effect.
• ANOVA repeated-measure test has more power than an ANOVA independent-measure analysis. It is more likely to detect a real treatment effect.

ANOVA repeated-Measures: Factors Influence

• hypothesis test outcome is influenced by the following three major factors:
  1. Size of the treatment effect :
     • • Bigger the treatment effect, the more likely it is to be significant.
       • larger treatment effect tends to produce a larger measure of effect size
  2. Size of the sample(s)
• A large sample effect is more substantial than an effect obtained with a small sample.
• Larger samples increase the likelihood of rejecting the null hypothesis.
• Sample size has little or no effect on measures of effect size
  1. The variance of the scores

Comparison: t-test vs. ANOVA repeated-measures

• • The two tests always reach the same conclusion about the null hypothesis
  • The basic relationship between the two test statistics is F = t ²
  • df value for the t statistic is identical to the df value for the denominator of the F-ratio.
  • The square of the critical value for the two-tailed t-test is equal to the critical value for the F -ratio. Relationship is F = t ²
  • F = t ²

Example for F = t ²

Refer following table Four participants have been given two treatments and the output is mentioned in the table.

• Examine that there is a significant difference in the treatment effects.
• Also verify F = t ²

Solution:

• calculate SS1, mean1 of treatment-I as per the below table :

• • • Mean1=6.6
    • SS1 = 5
    • n1 = 4

• calculate SS2, mean2 of treatment-II as per the below table :

• • • Mean2=9
    • SS2 =50
    • n2=4

• Calculate Sample Mean difference= MD

MD = Mean 2- Mean1

MD = 9 – 6.5 

MD = 2.5

• The sum of square difference = SSD

SSD = SS1 – SS2

SSD = 50 – 5

SSD = 45

• The sum of square within SS_within

• Degree of Freedom

df _{within treatment}   =  (n1 – 1) + (n2 – 1)

df _{within treatment}   =  (4 – 1) +  (4 – 1)

df _{within treatment}  = 6

• Sample variance

• Standard error:

• t value: calculation

• Calculate G & N 

G= 62

N=8

• Calculate sum of X*X

• Total Sum of square: SS_total 

• The sum of squares within SS_within 

• Calculate P (P = I + II)

• The sum of squares between participants:

• The sum of square Error:

• Degree of freedom -error: df_error 

df_error= (k – 1)(n – 1)

df_error= (2– 1)(4 – 1)

df_error = 1 * 3 = 3

k- no of treatment, n=no of participants

• MS and F-Ratio

• Relation between F-Ratio and T value:

F-ratio and the t statistic are related by the equation F = t ²

F= 1.66 = 1.29² 

t = 1.29

t ² = 1.29² 

t ² = 1.66 = F

Table-1 : F-Ratio Table with Alpha = 0.05

Table-2 : F-Ratio Table with Alpha = 0.01

Table-3 Studentized Range Statistic(q) at alpha= 0.05

Table-4 Studentized Range Statistic(q) at alpha= 0.05

Also Read

• https://matistics.com/statistics-data-variables/
• https://matistics.com/descriptive-statistics/
• https://matistics.com/1-1-measurement-scale/
• https://matistics.com/point-biserial-correlation-and-biserial-correlation/
• https://matistics.com/2-0-statistics-distributions/
• https://matistics.com/1-2-statistics-population-and-sample/
• https://matistics.com/7-hypothesis-testing/
• https://matistics.com/8-errors-in-hypothesis-testing/
• https://matistics.com/9-one-tailed-hypothesis-test/
• https://matistics.com/10-statistical-power/
• https://matistics.com/11-t-statistics/
• https://matistics.com/12-hypothesis-t-test-one-sample/
• https://matistics.com/13-hypothesis-t-test-2-sample/
• https://matistics.com/14-t-test-for-two-related-samples/
• https://matistics.com/15-analysis-of-variance-anova-independent-measures/
• https://matistics.com/16-anova-repeated-measures/
• https://matistics.com/17-two-factor-anova-independent-measures/
• https://matistics.com/18-correlation/
• https://matistics.com/19-regression/
• https://matistics.com/20-chi-square-statistic/
• https://matistics.com/21-binomial-test/