Analysis of Variance
ANalysis Of VAriance (ANOVA)
ANOVA is a statistical analysis technique in which the total variation is divided into meaningful sources of variation.
Recall from Measurement System Analysis,
Analysis of Variance (ANOVA)
ANOVA is often used for studying the relationship between a response variable (Y) and one or more explanatory or predictor variables (X’s). The predictor variables are also called factors or treatments.
While the response is quantitative, the predictors may be either quantitative or qualitative. However, quantitative predictors are analyzed as if they are qualitative (or categorical).
Analysis of Variance (ANOVA)
To study the effect of one or more factors on a response, each factor having two or more levels.
ANOVA can be used to
determine the statistical significance of effects
calculate the components of variance
estimate the contribution to variation by each identified source
estimate the underlying noise within the process
ANOVA – Principle
N(1,1²)
N(2,2²)
N(3,3²)
N(4,4²)
ANOVA - Terminology
yij is the jth observation when the factor is at its ith level.
y14
y23
y31
y42
ANOVA - Terminology
yi· is the expected response when the factor is at its ith level.
y1·
_
y2·
_
y3·
_
y4·
_
_
ANOVA - Terminology
y·· is the overall mean of all observations across all factor levels.
y··
_
_
_
_
ANOVA - Terminology
i = yi·– y·· is the effect of the factor at its ith level.
y··
_
_
1
2
3
4
_
_
_
ANOVA - Terminology
ij = yij – yi· is the residual or error of the jth observation at the ith level of the factor.
y··
_
_
14
24
31
43
_
ANOVA - Terminology
y··
_
_
Total Deviation
Unexplained Deviation
Explained Deviation
Total Deviation
= yij – y··
_
_
Unexplained Deviation
= yij – yi·
_
Explained Deviation
= yi· – y··
_
_
_
Total Deviation
= Explained Deviation
+ Unexplained Deviation
ANOVA table
1) DFTotal = DFFactor + DFError
DFError = DFTotal – DFFactor
= (N – 1) – (k – 1) = N – k
2) SSTotal = SSFactor + SSError
ANOVA Table
Total Deviation = Explained Deviation + Unexplained Deviation
SSTotal = SSExplained + SSUnexplained = SSFactor + SSError
Hypothesis Testing
H0 : 1 = 2 = = k
all group means are equal
Ha : i j for some i j
at least one pair of group means is not equal
ANOVA verifies the null hypothesis by comparing the variance between the groups against the variation within a group mean:
The null hypothesis is rejected if .
Hypothesis Testing
In general,
As will be seen in following examples, the actual formulation of F* is dependent on the type of ANOVA model.
ANOVA - Assumptions
The observations are mutually independent.
Stat Nonparametrics Runs Test
The k groups exhibit homogeneity of variance.
. 1² = 2² = = k²
Stat ANOVA Test for Equal Variances
The residuals are normally distributed.
. e ~ NID(0,²)
Stat Basic Statistics Normality Test
ANOVA - Assumptions
ANOVA - Models
Fixed Effects (ANOVA I) Model
factor levels in the experiment are specifically chosen
the conclusion is relevant only for the chosen levels and may not be extended to similar treatments that were not considered
Random Effects (ANOVA II) Model
factor levels are randomly drawn from a large population of treatments
the conclusion is valid for the entire population of treatments
Mixed Effects (ANOVA III) Model
one or more factors are fixed factors while others are random factors
ANOVA - Models
A k–Way ANOVA model involves the study of k factors.
k–Way ANOVA I : k factors; all factors of fixed effect
k–Way ANOVA II : k factors; all factors of random effect
(Will not be discussed in GB course)
k–Way ANOVA III : k factors; some factors of fixed effect
(Will not be discussed in GB course) others of random effect
One-Way ANOVA-I Model
where Yij is the value of the response variable in the jth trial
for the ith factor level or treatment
. is the overall mean for all observations
i=i–. is the effect of the ith treatment .
ij is the random error component .
The hypothesis may now be rephrased:
H0 : 1 = 2 = = r = 0
Ha : not all i equal 0
One-Way ANOVA-I Model
Example
An engineer want to evaluate the effect of the percentage of the strengthener of the adhesive to the bonding force (breakage force of the assy) of a bonded actuator.
The engineer decides to investigate four levels of percentage of the strengthener (ie, 5%, 10% 15% & 20%). Six samples were tested at each percentage level.
The data is stored in “4. ”
Example
Stat ANOVA One-way
Example
Session Windows
One-way ANOVA: Force versus Strengther
Analysis of Variance for Force
Source DF SS MS F P
Strength 3
Error 20
Total 23
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev -----+---------+---------+---------+-
5 6 (---*---)
10 6 (---*----)
15 6 (---*---)
20 6 (---*----)
-----+---------+---------+---------+-
Pooled StDev =
Example
Example
Example
Stat Regression Residual Plots
Example
One-Way ANOVA-II Model
where Yij is the value of the response variable in the jth trial
for the ith factor level or treatment
. is the overall mean for all observations
i=i–. is the effect of the ith factor level .
ij is the random error component .
The hypothesis may now be rephrased:
H0 : ² = 0
Ha : ² > 0
One-Way ANOVA-II Model
There is no difference in calculation of the F-Statistic between a One-Way ANOVA-I Model and a One-Way ANOVA-II Model . The difference lies in the conclusion.
Example
An engineer is interested in winding machine variability in peel strength of a coil. The engineer selects four winding machines at random and determines the peel strength of four samples chosen at random from each winding machine.
Example
Stat ANOVA General Linear Model
Example
Session Windows
General Linear Model: Peel Strength versus Machine
Factor Type Levels Values
Machine random 4 1 2 3 4
Analysis of Variance for Peel Str, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Machine 3
Error 12
Total 15
Example
End of Presentation
Rev 1 25 Jun 02
Rev 1: first release
