Lecture 6: ANCOVA Flashcards
In ANOVA (RECAP) we compare several means without increasing
the chance of type 1 error (familywise error)
Assumptions of ANOVA RECAP - (4)
Independence of data
DV is continuous (interval or ratio); IV categorical (3 groups)
No significant outliers;
DV approximately normally distributed for each category of the IV
When p < 0.05 in ANOVA (RECAP) there is a significant difference between
two or more of the means.
What is the decision tree for choosing ANCOVA? - (5)
Q: What sort of measurement? A: Continuous
Q:How many predictor variables? A: Two
Q: What type of predictor variable? A: Both Categorical and Continuous
Q: How many levels of the categorical predictor? A: Two or more
Q: Same or Different participants for each predictor level? A: Assumed different
one-way ANOVA could be charactercised in terms of a
multiple regression equation that used dummy variables to code group memberships
regression equation which includes a lot of predictor variables for ANOVA can extended to include one or more
continous variables that predict outcome/DV
regression equation for ANOVA
can be extended to include one or more continuous variables that predict the outcome (or dependent variable).
these continous variables are not
part of the main experimental manipulation but have an influence on the dependent variable, are known as covariates and they can be included
in an ANOVA analysis.
ANCOVA is an extension of multiple regression as it alllows you to
test all the regression lines to see which have different intercepts (B0) as long as all your slopes are equal (Same B1)
What does ANCOVA involve?
When we measure covariates and include them in an
analysis of variance
ANCOVA is extension of ANOVA as - (2)
- Control for Covariances (continuous variables you may not necessarily want to measure)
- Study combinations of categorical and continuous variables – covariate becomes the variable of interest rather than the one you control
Continuous variables, that are not part of the main experimental manipulation (don’t want to study them) but have an influence on the dependent variable, are known as
covariates
What does ANCOVA stand for?
Analysis of covariance
From what we know from hierarchical regression model, if we enter covariate into regression model first then dummy variables representing exp manipulation after… - (2)
then we can see what effect an IV has after the effect of covariate
We partial out the effect of covariate
What are the two reasons for including covariates in ANOVA? - (2)
- To reduce within-group error variance = if we can explain unexplained variance , SSR, in terms of other variables (covariates)then reduce SSR to accurately assess effects of SSM
- Elimination of confoundd = remove bias of unmeasured variables that confound results and influence DV
ANCOVOA has same assumptions of ANOVA, e.g., normality and homogenity of variance (Levene’s test) expect has two more important assumptions which are… - (2)
- Independence of the covariate and treatment effect
- Homogeneity of regression slopes
For ANCOVA to reduce within-group variance by allowing the covariate to explain some of the error variance the covariate must be
independent from the experimental/treatment effect - (IVs - categorical predictors) ( ANCOVA assumption)
What does independence of covariate mean in ANCOVA?
Independence of the covariate and treatment effect means that the categorical predictors and the covariate should not be dependent on each other
People should no use ANCOVA when the effect of covariate overlaps with the experimental effect as it means the
experimental effect is confounded with the effect of covariate = interpretation of ANCOVA is compromised
Similar issue in multiple regression - if predictor variable correlate too much with each other, it is
unlikely that a reliable estimate of their relationship with the outcome can be calculated
In ANCOVA, the effect of the covariate should be independent of the
experimental effect
When an ANCOVA is conducted we look at the overall relationship between DV and covariate meaning we fit a regression line to
entire dataset and ignore which groups pps fit in
In ANCOVA, we fit an overall regression line to the entire data set, ignoring to which group a person belongs.
In fitting this overall model we, therefore, assume that this
overall relationship is true for all grps of pps (e.g., if there is a positive relation between covariate and outcome in one group assume positive relation in all grps) - homogenity of regression slopes
What does homogenity of regression slopes mean in ANCOVA?
Homogeneity of regression slopes means that the covariate has a similar relationship with the outcome measure, irrespective of the level of the categorical variable - in this case the group
When is homogenity of regression slope is not satisifed in ANCOVA?
the relationship between the
outcome (dependent variable) and covariate differs across the groups then the overall regression model is inaccurate (it does not represent all of the groups).
What is best way to test homeogenity of regression slopes assumption in ANCOVA?
imagine plotting a scatterplot for each experimental condition with the covariate on one axis and the outcome on the other and calculate its regression line
Diagram of regression slopes satisfying homogenity of regression slopes in ANCOVA
- exhibits the same slopes for control and 15 minute group
Diagram of a regression slopes not satisfying homogenity of regression slopes in ANOCVA
- 30 minutes of therapy exhibts a different slope compared to others
What is design, variables and test would you use to test this researh scenario? - (5)
- ANCOVA
- Independent samples-design
- One IV , two conditions, interval regime and steady state
- One covariate (age in years)
- One DV (Race time)
What does SPSS output show? - (3)
- Interval group has race time of 57 minutes
- Steady state has race time of 62.97 minutes
- Levene’s test shows we have equal variances since we have a non-significant p-value (p = 0.934)
What does this ANCOVA output show?
- IV = Regime –> steady or interval
- Covariate = Age
- DV = Racetime- (2)
- Age F(1,27) = 5.36, p = 0.028, partial eta-squared = 0.17 (large and sig main effect)
- Regime F(1,27) = 4.28, p = 0.048, partial eta-squared = 0.14 (large and sig main effect)
What DF do you report from this ANCOVA table for age for example…
DF for age and DF for error
What does this SPSS output for ANCOVA show? - (3)
- Interval has a marginal mean of race times of 56.57
- Steady state has a marginal mean of race times 62.97
- Estimated marginal means partialled out the effects of age and view mean scores of race times in interval and steady state if mean age scores (30.07) across two groups was held constant
What does this output show in terms of homogenity of regression slopes?
age is covariate and regime is IV and DV is race times - (2)
- Interaction effect of regime * age has a p-value of 0.980
- Since p-value is not significant the assumption of homogeneity of regression slopes has been met
What does this SPSS output show in terms of independence of covariate and exp effect (IV)?
age is covariate (treated as DV) , regime is IV - (2)
- P-value is not signifcant (p=0.528) so effect of variable age is not sig difference of age across training regime
- and so independent variable are assumed to be independent.
We can look at parameter estimates table in ANCOVA to
interpret the covariate
What does positive and negative b-value for covariates in ANCOVA parameter estimate box indicate? - (2)
f the b-value for the covariate is positive then it means that the covariate and the outcome variable have a positive relationship
If the b-value is negative it means the opposite: that the covariate and the outcome variable have a negative relationship
What does this table of parameter estimates show for ANCOVA where..
DV = PP’sLibido, IV = Dose of Viagara, Covariate is Partner’sLibido - (3)
- b for covariate is 0.416
- Besides other things being equal, if a a partner’s libido increases by one unit, then the person’s libido should
increase by just 0.416 units - Since b is positive then partner’s libido ahs pos relation with pps’s libido
How is DF calculated for these t-tests in ANCOVA table? - (2)
N - p -1
N is total sample size, p is number of predictors (2 dummy variables and covariate )
What does these planned contrast results show in ANCOVA?
DV = Pp’s Libido, IV = Dose of Viagara, Covariate is Partner’s Libido -
IV Dose: Level 3 = high dose, level 2 = low dose, level 1 = placebo
(3)
- Contrast 1 of comparing level 2 (low dose) against level 1 (placebo) is significant (p = 0.045)
- Contrast 2 of comparing level 3 (high dose) with level 1 (placebo) is significant (p - 0.010)
What does this Sidak correction post-hoc comparison in ANCOVA output show?
DV = Libido, IV = Dose of Viagara, Covariate is Libido -
IV Dose: Level 3 = high dose, level 2 = low dose, level 1 = placebo
- (3)
- The significant difference between the high-dose and placebo groups remains (p = .030)
- high-dose and low-dose groups do not significantly differ (p = .93)
- Low dose and placebo groups do not significantly differ (p value = 0.130)
What do these scatterplot of regression lines show in terms of homogenity of regression slopes?
DV = Libido, IV = Dose of Viagara, Covariate is Libido -
IV Dose: Level 3 = high dose, level 2 = low dose, level 1 = placebo
(3)
For placebo and low dose there appears to be a positive relationship between pp’s libido and that of their partner
However, in the high-dose condition there appears to be no relationship at all between participant’s libido and that of their partner - shows negative relationship
Doubts whether homogenity of regression slopes is satisfied as not all the slopes are the same (go same direction)
How is eta-squared calcuated?
Dividing the effect of interest SSM by total variance in the data SST
How is partial eta-squared calculated for ANCOVA??
SS Effect/ SS Effect + SS Residual
What test is used to investigate this question and how is it conducted? - (2)
We want to know whether or not studying technique (3 levels) has an impact on exam scores,but we want to account for the grade that the student already has in the class.
- ANCOVA
- ANCOVA is conducted to determine i f there is a statistically significant difference between different studying techniques (IV) on exam score (DV) after controlling for current grade (covariate)
What ANCOVA was conducted?
We want to know whether or not studying technique has an impact on exam scores,but we want to account for the grade that the student already has in the class.
A three-way ANCOVA was conducted to determine a statistically significant difference between different study techniques on students exam scores after controlling for their current grades.
In ANCOVA, we partion the total variance into
IV, DV and covariate
In one-way ANOVA we partition the total variance into
IV and DV
Diagram of how to analyze ANCOVA on SPSS - (3)
outcome/DV = happiness measure ranging from 0 to 10 (as happy as I can image) = continous = interval
The fixed factor (IV) which is dose of therapy which is people have 15 minutes of puppy therapy or 30 minutes
Covariate is control group is how much they love puppies = continous = interval
Between subject effects output for ANCOVA is important as… - (2)
outcome/DV = happiness measure ranging from 0 to 10 (as happy as I can image) = continous = interval
The fixed factor (IV) which is dose of therapy which is people have 15 minutes of puppy therapy or 30 minutes
Covariate is control group is how much they love puppies = continous = interval
Rows 3 and 4 of the table look at the effects of the the covariate, puppy love, and the predictor, therapy dose, respectively.
The F and p values can be read from the final two columns
In ANCOVA, between subject effects we quote DF such as for dose as…
Quote df for the effect and error, e.g. 2,26
In ANCOVA, adjusted means table in SPSS shows.. - (2)
outcome/DV = happiness measure ranging from 0 to 10 (as happy as I can image) = continous = interval
The fixed factor (IV) which is dose of therapy which is people have 15 minutes of puppy therapy or 30 minutes
Covariate is control group is how much they love puppies = continous = interval
The group means can be recalculated once the effect of the covariate is ‘discounted’ = impact of covariate is taken into account and adjusted into each level of predictor variable in mean column
These values can differ markedly from the original group means and help with interpretation.
A psychologist was interested in the effects of different fear information on children’s beliefs about an animal. Three groups of children were shown a picture of an animal that they had never seen before (a quoll). Then one group was told a negative story (in which the quoll is described as a vicious, disease-ridden bundle of nastiness that eats children’s brains), one group a positive story (in which the quoll is described as a harmless, docile creature who likes nothing more than to be stroked), and a final group weren’t told a story at all. After the story children rated how scared they would be if they met a quoll, on a scale ranging from 1 (not at all scared) to 5 (very scared indeed). To account for the natural anxiousness of each child, a questionnaire measure of trait anxiety was given to the children and used in the analysis
what analysis has been used -
Independent analysis of variance
Repeated-measures analysis of variance
Mixed analysis of variance
Analysis of covariance
Analysis of covariance (ANCOVA)
A psychologist was interested in the effects of different fear information on children’s beliefs about an animal. Three groups of children were shown a picture of an animal that they had never seen before (a quoll). Then one group was told a negative story (in which the quoll is described as a vicious, disease-ridden bundle of nastiness that eats children’s brains), one group a positive story (in which the quoll is described as a harmless, docile creature who likes nothing more than to be stroked), and a final group weren’t told a story at all. After the story children rated how scared they would be if they met a quoll, on a scale ranging from 1 (not at all scared) to 5 (very scared indeed). To account for the natural anxiousness of each child, a questionnaire measure of trait anxiety was given to the children and used in the analysis
what is covariate?
Natural Fear Level
