Chapter 10-The Analysis of Variance
What are the assumptions of an ANOVA?
1) The samples are independent and identically distributed. -samples size must be equal within groups - assume the data represent a random sample 2) Variances are homogenous among groups - although means may differ from one another, we assume that the variance within each group is approximately equal to the variance within all the other groups 3) The residuals are normally distributed. - residuals are assumed to follow normal distribution with a mean equal to zero 4) The samples are classified correctly. - each y observation is put in correct treatment (pg 296) 5) The main effects are additive. - sometimes it is necessary to assume that the main effects are strictly additive, and there are no interactions among different factors.
If there is no interaction in a two-way what does this mean conceptually? What can we do at this point?
A and B are individual populations and do not depend on eachother At this point, we can interpret individual factors
What is an analysis of variance (ANOVA)?
A method of partitioning the sum of squares, due to Fisher. It is used primarily to test the statistical null hypotheses that distinct treatments have no effects on measured response variables.
A subscripted variable that does not have a bar over it indicates what?
A particular datum, such as Yij
What does a non-significant p-value tell us?
Accept null: all means are equal to the grand mean (p>0.05)
How can you prevent violation of the fourth assumption which states "Samples are classified correctly"?
Can only be complied by design
What does a block control for?
Confounding variable
If the within sum of squares is large relative to the sum of squares among groups what can we conclude?
Differences among the groups are weak or inconsistent.
What does a negative Ai term indicate?
Expectation is below the grand average.
What p-value should you interpret first in a two way ANOVA table?
Factor A*B
T/F the components of variation are not additive.
False! The components of variation ARE additive. SStotal=(SSamong groups)+ (SS within groups)
What is the equation for a two way ANOVA model?
Figure 10.10 pg 306
What is the equation for a two way ANOVA and what makes it unique?
Figure 10.10 pg 306 Interaction between two factors (Interaction effect)
What is the equation for SS among and explain how you would calculate it? Talk about equivalent equation you can use for this and what symbols mean.
Figure 10.2 pg 292, Equivalent equation is on pg 292 Take each treatment mean, subtract it from the grand mean, square it, and add the terms up for each of the a treatment groups n=number of trials per treatment y bar i = each mean per treatment y bar= grand mean; mean of all the data Tells us how far away each individual treatment is from grand mean Go over Example on pg 293
What is the equation for SS within,explain how you would calculate it?
Figure 10.3 pg 293 Calculate the deviation of each observation from its own group mean and then sum across the groups and the replicates. Go over Example on pg 293
If the ANOVA assumptions are met( or not too severely violated), we can test hypotheses based on an underlying model that is fit to the data. For the one way ANOVA, that model is represented by what equation?
Figure 10.6 pg 297 Yij= the replicate j associated with treatment level i miu= true grand mean or average Eij= error term Ai= represents the additive linear component associated with level i of treatment A; there is a different coefficient Ai associated with each of the i treatment levels.; represent deviations from grand means
If there are no treatment affects, then Ai=0 for all treatment levels. Therefore the null hypothesis is?
Figure 10.7 pg 297 if the null hypothesis is true than any variation that occurs among the treatment groups will reflect random error and nothing else.
What is the equation for a block ANOVA model?
Figure 10.8 pg 301
What are the two sources of the total sum of squares and what do they mean?
First Component= Variation AMONG groups -represents differences among the means or averages of each of the treatment groups.
What does a statistical analysis involve?
Involves specifying an underlying model for how observations might be affected by different treatments, partitioning the sum of squares among the different components in he model, then using the results to test statistical hypotheses for the strength of particular effects.
If differences among blocks are large, the reduction in the among-group sum of squares will be ___________.
Large Test for treatment effect more powerful
Which test can we use to ensure that variance is homogenous among groups?
Levenne's Test
What does a lower case letter of the factor symbol mean?
Maximum number of levels for a factor is indicated by the corresponding lower case of the letter. e.g. if Ai indicates treatment level i for Factor A, the number of levels of Factor A ranges from i=1 to a. Exception: lower case n
What does the symbol Y mean?
Measured response variable
What are the two hypotheses tested for a block ANOVA?
Null Hypothesis: no differences among blocks -F ratio can test this Second null hypothesis: no differences among treatments
What are the hypotheses for a Levenne's test and what is the cutoff?
Null(H0)= variances are equal Ha= variances are not equal x>= 0.01 Accept the null x<0.01 Reject null
How many p-values do you get in a one way ANOVA table and what is it for?
One for the source of variation
If you get a significant result what test should you do to compare every possible combination (p-values)?
Post-Hoc Tukey Test will tell you exactly which population is different
In more complex ANOVA models we usually partition the total sum of squares into multiple components of variation each representing the contribution of a factor in the model. In all cases,though, what is left over is always the ___________.
RSS (residual sum of squares)
What are the hypotheses for a two way ANOVA?
Reject null (p<=0.05) : Interaction Accept null (p>0.05): No interaction
What does a significant p-value indicate?
Reject null:That there is a significant difference in at least one of the means. (p<=0.05) reject null of no treatment effect
The within group sum of squares is often called the __________.
Residual sum of squares, Residual variation, or error variation As in a regression, we refer to it as the residual because it is the variation that is not explained by controlled or experimental factors of our model. Our statistical model incorporates this component as random sampling from normal distribution.
Which SS should be higher in an ANOVA table?
SS among
Which equation accounts for the remaining component of variation withing groups?
SS within
What test can we use to test the assumption that residuals are normally distributed?
Shapiro Wilks H0= Residuals are normally distributed Ha= Residuals are NOT normally distributed p>= 0.01 Accept H0 p<0.01 Reject H0
If the differences among blocks are small, the reduction in the among group sum of squares will be ________-.
Small Test for treatment effect less powerful
What are the types of components that can contribute to variation?
Some components can represent random or error variation that is not attributable to any specific cause; it may result from observation error and other unspecified sources. Other components represent the effect of the experimental treatments applied to the replicates or the differences among sampling categories.
The analysis of variance is built on the concept of partitioning the _______ _________ ________.
Sum of squares The total variation can be partitioned or divided into different components
In the one-way ANOVA model, the controlled factors represent processes that we hypothesize cause differences among the treatment groups. The effect of these factors is represented by the _________ _____ _______ _____ _____.
Sum of squares among groups;
If the sum of squares among groups is relatively large compared to the sum of squares within groups what does this tell us?
That differences among treatments would seem to be important
What is the difference between the equation for a one way and the equation for a block?
The block equation now has a block effect Bj contains sum of squares for differences among treatment means but also contains a sum of squares for the differences among blocks.
What does the total sum of squares represent?
This total sum of squares reflects the deviation of each observation the grand mean. It can be decomposed (partitioned) into two different sources.
How many p-values do you get in a block ANOVA table and what are they for?
Three (Factor A, FactorB, and Factor A*B)
Sum of squares partitioning is a natural measure of ____________ effects.
Treatment
What does a positive Ai term indicate?
Treatment level i has an expectation that is greater than the grand average.
How many p-values do you get in a block ANOVA table and what are they for?
Two ( one for Among and one for block)
Do we want SSamong to be larger or SS within and why?
We want SS among to be larger so that we can say that treatments were individual populations.
Where should we have the least SS in block ANOVA table?
Within
What do capital letters A,B, and C mean?
designate the different factors in the model
What does the interaction effect measure?
differences in the means of treatment groups that cannot be predicted on the additive basis of the two main effects
What do the subscripts i,j,k..... indicate?
different levels of variables Treatments of the factors e.g. we would write Ai to indicate level i of Factor A, and Bj to indicate level j of Factor B.
What does the symbol Y bar mean?
indicates the grand mean of the data
What does the symbol n mean?
number of replicates used to estimate the sum of squares withing groups (the residual sum of squares), the lowest level at which replicate samples are taken. number of replicates/treatment
What is the equation for the total sum of squares?
pg 291 Figure 10.1
What does the variable miu indicate?
the expected value of a variable in the model
What does the symbol E mean?
the residual error term is indicated by E, usually with some subscripting to indicate the different treatment components.
What types of variables are used in an ANOVA (continuous/categorical)?
x(predictor variable)= categorical y(response variable)=continuous