Analysis of (Co-)Variance
Analysis of Variance (ANOVA): is a statistical test to determine whether there are
significant di2erences among the means of two or more groups. It assesses whether the
variability between group means is larger than the variability within the groups.
è The null hypothesis (𝐻! ) states that all group means are equal
è The dependent variable in ANOVA must be a metric variable. This means it must be
measurable on an interval scale, or a ratio scale.
è ANOVA requires at least one or more independent variables, called factors, these
factors must be categorical (non-metric). Each level of a factor represents a
di2erent category for comparison.
How does ANOVA work:
• ANOVA divides the total variance observed in the data into between-group
variance (di2erences due to the independent variable )and within-group variance
(di2erences due to randomness or noise).
o If the between-group variance is significantly greater than the within-group
variance, the test concludes that the group means are not all equal, rejecting
the null hypothesis.
Three di2erent types of ANOVA:
1. One way ANOVA: This involves one categorical independent variable (or one factor)
2. N-way ANOVA: This occurs when two or more factors are involved as independent
variables.
3. ANCOVA: Is a more advanced form of ANOVA that includes both categorial and
metric independent variables. The categorical variables are still treated as factors,
while the metric independent variables are referred to as covariates.
, One Way ANOVA: Examines di2erences in the mean of a dependent variable across
categories of a single independent variable (factor)
Applications:
o Whether the number of defective products di2er across di2erent machines
o If work-related accidents vary across di2erent working environments
o How di2erent troubleshooting checklists a2ect repair times.
Step by step calculation
1. Calculate the average of the dependent variable across all observations.
"#$%& #( $)* +*,*-+*-$ .%/0%1&*
§ 𝑌# = "#$%& -231*/ #( #14*/.%$0#-4
2. Calculate the Sum of Squares (𝑆𝑆5 )
§ 𝑆𝑆5 = ∑(𝑌0 − 𝑌#)6
3. Calculate between-group sum of squares (𝑆𝑆7 ) (measures the variability by
di2erences in group means)
§ 𝑆𝑆7 = ∑ 𝑛𝑗(𝑌#8 − 𝑌#)6
- nj = number of observations in group
- 𝑌"! = mean average of group j
- Calculate deviations for each group mean from the grand mean, square them and multiply
by group size
4. Calculate within-group Sum of Squares (𝑆𝑆*//#/ )
§ 𝑆𝑆*//#/ = ∑(𝑌08 − 𝑌#8 )6
- Here you subtract the average mean of the group of the actual value of the observation in
that group.
- So you take each observations value, substract their group mean, square the result and
sum everything in that group.
F-Test for significance: The F-test for significance compares the variance between groups
to the variance within groups:
9*%- 4:2%/* 1*$;**- </#2,4 (9>?) >>! / (BCD)
𝐹= 9*%- 4:2%/* ;0$)0- </#2,4 (9>A)
=𝐹 = >>$%%&%/ (ECB)
- Where N = total of observations
- C = number of observations in the group
A significant F-statistic suggests that the dependent variable is influenced by the
independent variable. The null hypothesis can then be rejected.
Analysis of Variance (ANOVA): is a statistical test to determine whether there are
significant di2erences among the means of two or more groups. It assesses whether the
variability between group means is larger than the variability within the groups.
è The null hypothesis (𝐻! ) states that all group means are equal
è The dependent variable in ANOVA must be a metric variable. This means it must be
measurable on an interval scale, or a ratio scale.
è ANOVA requires at least one or more independent variables, called factors, these
factors must be categorical (non-metric). Each level of a factor represents a
di2erent category for comparison.
How does ANOVA work:
• ANOVA divides the total variance observed in the data into between-group
variance (di2erences due to the independent variable )and within-group variance
(di2erences due to randomness or noise).
o If the between-group variance is significantly greater than the within-group
variance, the test concludes that the group means are not all equal, rejecting
the null hypothesis.
Three di2erent types of ANOVA:
1. One way ANOVA: This involves one categorical independent variable (or one factor)
2. N-way ANOVA: This occurs when two or more factors are involved as independent
variables.
3. ANCOVA: Is a more advanced form of ANOVA that includes both categorial and
metric independent variables. The categorical variables are still treated as factors,
while the metric independent variables are referred to as covariates.
, One Way ANOVA: Examines di2erences in the mean of a dependent variable across
categories of a single independent variable (factor)
Applications:
o Whether the number of defective products di2er across di2erent machines
o If work-related accidents vary across di2erent working environments
o How di2erent troubleshooting checklists a2ect repair times.
Step by step calculation
1. Calculate the average of the dependent variable across all observations.
"#$%& #( $)* +*,*-+*-$ .%/0%1&*
§ 𝑌# = "#$%& -231*/ #( #14*/.%$0#-4
2. Calculate the Sum of Squares (𝑆𝑆5 )
§ 𝑆𝑆5 = ∑(𝑌0 − 𝑌#)6
3. Calculate between-group sum of squares (𝑆𝑆7 ) (measures the variability by
di2erences in group means)
§ 𝑆𝑆7 = ∑ 𝑛𝑗(𝑌#8 − 𝑌#)6
- nj = number of observations in group
- 𝑌"! = mean average of group j
- Calculate deviations for each group mean from the grand mean, square them and multiply
by group size
4. Calculate within-group Sum of Squares (𝑆𝑆*//#/ )
§ 𝑆𝑆*//#/ = ∑(𝑌08 − 𝑌#8 )6
- Here you subtract the average mean of the group of the actual value of the observation in
that group.
- So you take each observations value, substract their group mean, square the result and
sum everything in that group.
F-Test for significance: The F-test for significance compares the variance between groups
to the variance within groups:
9*%- 4:2%/* 1*$;**- </#2,4 (9>?) >>! / (BCD)
𝐹= 9*%- 4:2%/* ;0$)0- </#2,4 (9>A)
=𝐹 = >>$%%&%/ (ECB)
- Where N = total of observations
- C = number of observations in the group
A significant F-statistic suggests that the dependent variable is influenced by the
independent variable. The null hypothesis can then be rejected.
