ANOVA Analysis of Variation

Analysis of Variance (ANOVA) is a parametric statistical technique used to compare data sets. This technique was invented by R.A. Fisher; hence, it is also referred to as Fisher’s ANOVA. It is similar to techniques such as t-test and z-test to compare means and also the relative variance between them.

Similarly, a t-test can be used to compare two sample means. What if we want to compare more than two means? Analysis of variance (ANOVA) is best applied where more than 2 populations or samples are meant to be compared.

It is used to test the statistical significance of the relationship between a dependent variable (“Y”) and single or multiple independent variables (“X’s”).

Types of ANOVA

• One-way
  • Measures single factor from multiple sources
  • Uses only one technician / one measurer
• Two-way (without replicates)
  • Measures 2 factors
  • Uses only one technician (unless technicians are one of the factors)
• Two-way (with replications)
  • Measures 2 factors but has multiple repetitions of each combination.
  • Uses only one technician (unless the technicians are one of the factors)

ANOVA Sum of Squares Correction Factor

• Grand total of all runs (G) = ΣX
• N= Total number of runs
• Correction factor (CF)= (ΣX)² /N = (G)² /N

Terms used in ANOVA

• Degrees of Freedom (df): The number of independent conclusions that can be drawn from the data. 
• SS_Factor: It measures the variation of each group’s mean to the overall mean across all groups. 
• SS_Error: It measures the variation of each observation within each factor level to the mean of the level. 
• Main effect: A main effect is an effect where the performance of one variable is considered in isolation by neglecting other variables in the study.
• Interaction: An interaction effect occurs where the effect of one variable is different across levels of one or more other variables.
• Mean Square Error (MSE): The mean square of the error (MSE) is divided by the sum of squares of the residual error by the degrees of freedom.
• F-test Statistic: The null hypothesis that the category means are equal in the population is tested by F Statistic based on the ratio of mean square related to X and mean square related to the error.
• P-value: It is the smallest level of significance that would lead to rejection of the null hypothesis (Ho). If α = 0.05 and the p-value ≤ 0.05, then reject the null hypothesis. Similarly if the p-value > 0.05, then fail to reject the null hypothesis.

https://youtu.be/x6F9uvaviEc

Analysis of Variance (ANOVA) has three types:

• One-way analysis
• Two-way analysis
• K-way analysis: K-Way ANOVA can be two-way ANOVA or three-way ANOVA, or multiple ANOVA

One way ANOVA

One-way ANOVA (one-way analysis of variance) is a statistical method to compare the means of two or more populations.

Assumptions of One-way ANOVA

• The sample data drawn from k populations are unbiased and representative.
• The data of k populations are continuous.
• The data of k populations are normally distributed.
• The variation within each factor or factor treatment combination is the same; hence, it is also called homogeneity of variance.
• Finally, the variances of k populations are equal.

Steps for Computing one-way ANOVA:

• Establish the hypotheses. H₀: µ1= µ2= µ3 and H₁: At least one group means differs from the others.
• In ANOVA, the total variance is subdivided into two independent variances: the variance due to the treatment and the variance due to random error.

• SS_T = SS_b + SS_w 
• Calculate the ANOVA table with degrees of freedom (df), and calculate the group’s error and total sum of squares.

• SS_b= sum of squares between treatments
• SS_w= sum of squares due to error
• MS_b= mean square for treatments
• MS_W= mean square for error
• SS_T= total sum of squares
• T= number of treatment levels
• n= number of runs at a particular level
• N= total number of runs
• F= the calculation of the F Statistic with k-1, and N-k is the degrees of freedom.
• Determine the critical value. F critical value from the F distribution table.
• Finally, Draw the statistical conclusion. If Fcalc< Fcritical, fail to reject the null hypothesis, and if Fcalc > Fcritical, reject the null hypothesis.

Example of One-Way ANOVA

A car manufacturer planned to conduct tests to determine the performance of 3 different brands of 12V batteries, so he selected five batteries from each brand and discharged them under controlled conditions. Assuming the lifetime of batteries is normally distributed at a 95% confidence level. The hypothesis is that the three brands have no difference in lifetime.

H₀: µ1= µ2= µ3

H₁: At least one of the brands mean life is different from the others.

k = 3, n = 5, N = 15, T1 = 71, T2 = 85, and T3 = 116, and G = 272 ΣX²= 1021 + 1489 + 2730 = 5240

Correction factor (CF) = (ΣX)² /N = (G)² /N = (272)²/15

One-Way ANOVA Excel Download

For α = .05, the critical value for F with d.f. (2, 12) is 3.89.

Fcalc > Fcritical; hence, we may have to reject the null hypothesis.

The calculated value does lie in the critical region. Therefore, there is evidence, at the 5% significance level, that the mean lifetime of the three brands’ batteries differs.

Example One-Way ANOVA

Three different brands of chlorine are used at a local pool throughout the summer. Initially, they used one brand. As one brand’s four-week supply runs out, the staff begins using the next brand of chlorine. Management is interested in determining if the different brands significantly affect the ability to maintain safe pH levels. Once the previous brand is no longer present in any measurable amount, the pH levels for each brand are collected through random sampling. Is there a significant difference between the three brands of chlorine? Test at the 5% significance level. Complete the ANOVA table (except the p-value) manually.

Two way ANOVA

Two-way ANOVA performs an analysis of variance for testing the equality of population means when the classification of treatments is by two categorical (independent) variables or factors.

Assumptions of Two-way ANOVA

• The cells contain independent samples
• Two main effects and an interaction
• The populations should have equal variance
• Balanced data and fixed factors

Steps for Computing Two-way ANOVA:

• Establish the hypotheses: The null hypotheses for each of the sets are given below:
• The population means of the first factor are equal. This is like the one-way ANOVA for the row factor.
• The population means of the second factor are equal. This is like the one-way ANOVA for the column factor.
• There is no interaction between the two factors. This is similar to performing a test for independence with contingency tables.
• Calculate the test statistic and rejection region.

• Calculate SS_{AB(Interaction)} as= SS_t-SS_w-SS_A-SS_B
  • a: number of levels in factor A
  • b: number of levels in factor B
  • r: total number of trials
  • Xi..: mean of the i th factor level of factor A
  • X…: overall mean of all observations
  • X_.j.: mean of the j th factor level of factor B
  • X_ij.: mean of observations at the i th level of factor A and the j th level of factor B
• Calculate the ANOVA table with degrees of freedom (df) and F value as shown below.

• Determine the critical value. Remember, F(critical) is from the F distribution table.
• Finally, Draw the statistical conclusion. If Fcalc< Fcritica,l fail to reject the null hypothesis, and if Fcalc > Fcritical, reject the null hypothesis.

Example:

A researcher experimented to check the effectiveness of the coating. A new coating was applied to two different materials, and research was conducted at two different laboratories. For instance, each laboratory tested five samples from each treated material. Find the results from the table below:

For α = .05, the critical value for F with d.f. (1, 16) is 4.49.

There appears to be both Laboratory and Material Fcalc > Fcritical; hence, we reject the null hypothesis. But for the interaction, Fcalc< Fcritical, hence, fails to reject the null hypothesis.

https://www.youtube.com/watch?v=7vLp8MDwgcE&feature=youtu.be

Multivariate Analysis of Variance (MANOVA)

Multivariate analysis of variance (MANOVA) is an ANOVA with several dependent variables. That is to say, ANOVA tests for the difference in means between two or more groups, while MANOVA tests for the difference in two or more vectors of means.

MANOVA examines the dependence relationship between dependent measures across a set of groups. Moreover, this analysis is used in experimental design, and the hypothesized relationship between dependent measures is usually used.

Assumptions of MANOVA

• The response variables are continuous.
• The residuals follow the multivariate-normal probability distribution with means equal to zero.
• The variance-covariance matrices of each group of residuals are equal.
• The individuals are independent.

Interestingly, in addition to detecting differences in the average values, it also detects differences in correlations among the dependent variables between the different levels of the independent variable.

Benefits of MANOVA

There are various advantages of MANOVA over ANOVA.

• Study any interaction between the factors.
• Study two or more factors simultaneously to increase the model’s efficiency.
• MANOVA reduces the chances of alpha risk.
• Less residual variation in the model when more factors are in the study.

Additional Resources for ANOVA

http://psychnut.com/statistics/even-more-about-1-way-anova/

https://www.technologynetworks.com/informatics/articles/one-way-vs-two-way-anova-definition-differences-assumptions-and-hypotheses-306553

http://oak.ucc.nau.edu/rh232/courses/EPS525/Handouts/Understanding%20the%20One-way%20ANOVA.pdf

Six Sigma Black Belt Certification ANOVA Questions:

Question: To assess the significance of factors in either a fractional or a full-factorial experiment structure, a black belt can be used: (Taken from ASQ sample Black Belt exam.)

(A) analysis of variance (ANOVA)
(B) fault tree analysis (FTA)
(C) failure mode and effects analysis (FMEA)
(D) evolutionary operation (EVOP)

Answer: 

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