week 5: ANOVA/ANCOVA Flashcards
what is ANOVA
“ANOVA” = ANalysis Of VAriance
a statistical technique to assess the differences between groups on a dependent variable (DV) or set of variables.
One or more dependent variables.eg, “Are, black, yellow,chocolate labradors different in relation to;
-life expectancy
-health problems
-cost of puppy
-popularity
-appeal? etc
ANOVA can also be used to assess differences between times or events.eg “is the behaviour of the same dogs similar or different over time?”
Or eg, same person “does mood change after tx”?.
Irrespective of ANOVA type, the independent variables are ALWAYS categorical (eg lab colour type, male/female, Australian/American/Asian, Time1/Time2, or pre-medication/post0medication etc.
The dependent variables are ALWAYS continuous.
Covariates may be continuous or binary.
Different types of ANOVA
different types of ANOVA include;
-between groups ANOVA
-within groups anova
-mixed anova
ancova
Between-groups ANOVA
Diferent subjects participate in each test condition. (testing for differences between different groups)
Within-groups ANOVA
(also called Repeated Measures)
The same subject participates in all conditions. Tests for individual differences across the conditions within the same grouo.
Mixed-design ANOVA
This is a combination of BETWEEN and WITHIN.
eg “Does change in dog behaviour over time depend on whether the dog is chocolate, yellow or black?” or eg “Does happiness increase after medication equally for men compared with women?
uni-variate anova
has 1 dependent variable only
multi-variate anova
has >1 dependent variable
eg Does health (as indicated by various health parameters) differ according to a dog’s colour? All DV’s must be correlated to each other to represent indicators of one parameter. ie set of health markers such as severity of skin rash, genetic markers and blood markers.The health markers correlate to each other to show the dv Health.
analysis of co-variance (ANCOVA)
Group differences are compared whilst controlling for another variable. eg Do chocolate labs have more skin probelms than yellow or black labs, controlling for age of dog?
The covariate should be significantly correlated with the dependent variable.(ie age of dog correlates with amount of skin dz). (ie b/c if were testing diff colour groups for skin issues, and 1 group happened to be a different age, you would want to be able to determine any difference in skin issues was due to the colour and not the age). When the covariate is significantly correlated with the dependent variable, we can remove the effect of differeng age groups by using regression, and removing age from the group mean. this has then allowed us to statistically consider just the coat colour and not the age, on skin issues in labs.
If there is one or more covariates, all designs remain the same. However, they become an analysis of covariance or ANCOVA—for example, a two-by-two mixed-design univariate ANCOVA or a three-by-four between-groups multivariate ANCOVA. The designs all stay the same, with the only difference being the added covariate, whether one or several.
Finally, when doing research, you typically need to include information on what sort of ANOVA or ANCOVA design you use. The following is an example of what that might look like.
Design write-up
A two-by-two (colour: black versus chocolate; food quality: low versus high) between-groups univariate ANCOVA was computed to test the hypothesis. The dependent variable was the dog’s health score, and the covariates were the dog’s age in years and the sex of the dog.
ANOVA design
Independent variable type (within or between) and numbers impact the design of an ANOVA. The ANOVA can have one or several independent variables (IVs), which can have two or more levels. When there is only one IV, it is called a one-way ANOVA, like in the following example where the RV is dog colours. This table also shows three levels, as there are three colours.
Between-groups one-way ANOVA
Dog colour DV = Mean health score
Chocolate 7.51
Black 8.01
Yellow 8.25
If there were only two levels—for example, only Chocolate and Black—it would still be a between-groups one-way ANOVA. When there are only two groups, it is the same as a t-test. below, is an example of a 2 way anova. there are 3 groups of dogs, and 2 groups of food. This means there are 6 possible conditions.
3 way anova
below is an eg of a 3 way anova. must bear in mind, that as consider more and more conditions, the greater the sample size must be to have significant power.
F test
F=between groups variance/within groups variance.
(variance b/n groups is considered to be error, because trying to argure groups are same and there is a difference between them.
F= between groups mean square(MS)/within groups mean square(MS)
Mean squares (MS) are calculated from the differences in individual dog scores from their colour group and from the grand total and then squared (sum of squares) and then divided by degrees of freedom.
degrees of freedom for the between groups MS (or effect) is number of groups -1. ie in this eg is 3-1=2.
degrees of freedom for the within groups MS (or error) is the total sample size -the number of groups. eg. 102-3=99.
If the hypothesis that there is a difference between the groups is NOT supported, the F statistic tends to be close to 1.
If the F statistic is a lot >1, then there is some evidence that there is a difference between the groups (and their means).
What are the degrees of freedom?
The number of values in a calculation that have the freedom to change or vary, given certain constraints or rules.
When we are calculating or estimating certain statistics, we will have formulas or rules on how to do this.
The formula is considered to be a constraint. eg beta1 + beta2=10.
Only 1 value here can vary. We could chose a range of numbers to be beta1 or beta2 etc but if we estimate beta1 to be 9, beta2 can only be 1. We cannot choose beta2. there is only 1 degree of freedom.
In the context of Anova, the Grand Mean of food quality must equal the average of food quality means across the groups.
eg Grand mean=((black lab food quality mean )+(yellow lab food quality mean)
+ (chocolate lab food quality mean))/3
eg if Grand mean =77.98 and chocolate mean=67, and black mean=99, then we know that yellow mean must equal 67.93. Ie if we choose values for chocolate and black, we can’t chose value of yelllow, as is set.
ie with 3 groups, we have 2 degrees of freedom.
one way between groups anova example
Research question
Does the health of a Labrador depend on its colour?
H1;Mean health scores will significantly differ across the three colours of Labradors.
H0;Mean health scores will not significantly differ across the three colours of Labradors.
dv=Continuous (quantitative) health score (Range: 10–100) where high scores = better health.
i/v =Categorical (qualitative) measure of the dog’s colour (three groups).
design=One-way between-groups ANOVA.
sample size=102 individuals (different dogs).
