Lecture 8: Analysis of Variance ANOVA Flashcards
ANOVA can be:
* One-Way
* Two-Way
* Repeated Measures
* Mixed Designs
How many means can a t test compare?
2
however, if you need to compare more than 2 means than you use an ANOVA
Anova can allow you to comapre 3+ means within the same group
* walking w/o a cane
* walking w/ a cane on R
* Walking w/ a cane on the L
So notice below were comparing the means all within 1 group
ANOVA can also allow you to compare 3+ group means in different groups
EX: - comparing means in 3 different groups
* Experimental group A = manual therapy
* Experimental group B = Manual therapy + EX
* Control group = standard care
Then we would compare the means and see if we can find a difference in them
Just like t-tests, ANOVAs assume data is parametric: Meaning
* Homogenetiy of variance (variance is about the same)
* Normal distribution - No skewing either direction
* Interval/ratio data - continuous data (equal distance between values) - like a MMT would not be one, it would be considered ordinal - something like gate speed would work (every second has an equal distance between it = continuous)
One-Way Anovia
* One factor being examined
The factor = the intervention
* We can have multiple levels of interventions, but the intervention itself is the only factor (that factor just has multiple levels)
In the example below we have 44 participants w/ elbow tendinits. They want to see out of these interventions, which one is the best. Is the best intervention Ice, Nsaids, Splint, or rest. Notice our factor here is intervention, and our levels are the 4 different kinds of interventions
Our outcome measure of pain free ROM was measured 10 days later to see which intervention was the best
Dependent variable = pain free ROM
Independent variable = intervention
H0 = u1 = u2 = u3 =4
* aka all the group means are the same, theres no difference between the means following the intervention
Ha = theres a significant difference between at least 2 of the groups
* any of these two groups are significant difference from the others, just saying something is not the same
The mean = the change from baseline following the intervention
* “On average the people who used ice improved roughly 44 degrees”
Standard deviation = the variability of that data
* all and all within 1 standard deviation people got better 44 degrees on average +/- 10 (ice group) - not the same as 95% confidence interval
* In an ANOVA our variability in the data should be fairly the same between tests (we did a special tests to make sure the variability in the data was not significantly different between the groups)
* 95% confidence interval means were 95% confident the true mean falls between those 2 points
On the right its plotting the means of the 4 different groups following the intervention (not over time) compared to the mean change in ROM. Can see that Nsaid performed best
* theres no order these are graphed its just showing the data
* visually speaking they look like theres a statistically significant difference (the most between Nsaid and rest)
For these we need to perform a test of homogeneity of variances (just like we did in t-tests) to make sure the variances between groups are the same
On the top chart we find a p value of .8 which is higher than our a of .05 meaning that it is not satistically significant, meaning the we equal variances between the group means.
* This tells us we can do our ANOVA
* This is the chart at the bottom
If there is a significant difference between the variance of the groups performing an ANVOA is not appropriate
So she would expect us to be able to say that they found a significant difference between groups (p < a) with a power of .999 (meaning theres a 99% chance that we correctly idientied a difference between groups) and we have a large effect size
So this is graphing the mean change (post intervention of each group), and what the variance is assumed to be (were assuming = variance, meaning the humps of roughly the same width)
Test of homogeneity of variances is done to see if the variances are the same. If its statsitically significant than the variance is different, however, if theres no signficiant difference than the variability between data sets is roughly =
This is still looking at the effect of different interventions on pain-free ROM
The first line wants to see if there is a differnce between groups.
* ignore the within groups line
F = a number that needs to be compared to something
p = 0.000, meaning that between the 4 intervention groups theres a difference somewhere
Observerd power
* the chance of correctly identifying a change between groups (there was a change and we correctly identified it) - the proabibility that we correclty identify a difference. It backs up our p value
* B = chance of type II error
* power = .9999 = great
Partial Eta Squared
* Represents our effect size
* .475 = large
Partial Eta Squared is the same thing as effect size (d) except its used in ANOVAs
* it also has a slightly different scale than the effect size used in t tests
**Small effect size (eta squared) =
Medium effect size (eta squared) =
Large effect size (eta squared) = **
Small = 0.01-0.06
Medium = 0.06-0.14
Large = 0.14+
A one-way ANOVA was run to compare the effect of ice, NSAIDs
A one way ANOVA was run to compare the effect of ice, NSAIDs, splint, or rest on change in pain-free ROM. There was a significant difference among the four modality groups (F (3,40 - these are df) = 12.06, p < 0.001, f = .951)
* One-way ANOVA = 1 factor being examined (intervention), however it has 4 levels
* p < 0.001 = significant difference between at least one of the group means from the others
* Independent variables = intervention
* Dependent variable = pain free ROM
* “there was a significnt difference among the four modaility groups” - means between 1 or more of the groups theres a significant difference in group means.
F (3,40) = 12.06
* these are df
* 3 represents the # of groups (n-1 = 4-1 =3)
* 40 = # of participants - # of groups = 44-4 =40
* based on these two variables our F = 12.06 - however, w/o a critical value to compare it to it means nothing
* through this f they ran the ANOVA and found the p value = 0.001, meaning there is a significant difference somewhere between at least one and another groups group mean
* f = 0.951 is not super useful to us. Its a way to document the effect size, but shes going to be giving us partial eta squared (it would be a different # once converted into partial eta squared), it would actaully = .475 (grabbed from example above)
So we now know theres a significant difference between 1 or more of the group means. However, how is this useful? it doesnt tell me where that siginifcant difference is.
If we want to figure out where the difference lies, we need to do a multiple comparesion test, it tells us where that exact difference between those group means lies
So the question would be why don’t we just run the multiple comparsion test right off that bat instead of doing the ANOVA at all? Well the Anova is much easier to run and more simple, its a good screening test, if we don’t find a difference in group means w/ the ANOVA than theres no point in running the multiple comparison test, however, if we find a difference in group means w/ our ANOVA analysis then we can implement this more time consuming multiple comparsion test to find where that exact difference in group means lies.
The ANOVA is an efficient way to determine if the null hypothesis should not be rejected
* basically if we find no difference in group means we keep the null hypothesis and theres no point in running the multiple comparison test
2-Way ANOVA means theres two factors
Factor 1 = Medication Intervention
Factor 2 = Conservative Intervention
Here there is 6 possible treatment combination (3x2)
2 factprs
Conservative Interventions (3 levels)
* Ice
* Splint
* Rest
Medication (2 levels)
* Nsaids
* No Nsaids
This would be considered a 3x2
In the picture below the top 3 are combinaing the conservative intervention w/ Nsaids. The bottom 3 are combining the 3 conserative treatment itnervention w/ no Nsaids (they just didnt state no nsaids, they just left that part off)
We want to see if theres any difference between the groups.
We also can find an interaction effect, meaning some magic combination of groups changes the outcome more
* some magic combination over the other 5 scenarios
Two-Way ANOVA
Askign 3 main questions
1) What is the effect of the modality, indepependent of medication?
* Main effect of modality (comparing between the different modalities independent of medication)
2) What is the effect of medication independent of modality
* Main effect of medication (comparing between medication vs no medication independent of modalities)
3) What is the combined effect/interaction of modality and medications
* Is there an interaction effect? - one combination different from the rest - aka 1 of these groups (or more) is statistically significant from 1 of the other groups
Make sure to know the difference between main effect (effect of one 1 factor independet of the other) and interaction effect (some combined magic effect)
So to find if there is a statsitically significant difference in the main effect of modality (independent of medication) you have to compare the group means between the 3 modality interventions (they’re all independent of medication)
In this example there are 60 people w/ elbow tendinitis and were looking at the effects of ice, splint, rest, and then those 3 combined w/ nsaids (6 total groups)
Below in the white chart were looking at their average improvement. For example for the people that used Ice+Nsaids they improved 45 degrees
* EX: The group that utilized both splint and NSAIDs improved pain free ROM 33 degrees
Marginal Means = the average independent of the other factor
* EX: On average those who utilized Nsaids improved 33.33 degrees
* Those who used ice (independent of medication) improved 40.50 degrees
The graphs are basically saying the exact same thing. on the first one you can see Ice+Nsaid = best combination (same thing on the second one)
* The trend seems to be that adding in medication increases pain free ROM more
You can see that the pattern of response is similar, but there is no magic combination
Main effect of medication would be that Nsaids always did better than no medication
Main effect of modality says regardless of Meds/No meds ice was always the best
Modality line: tells us the main effect of the modality
* is there a significant difference between the groups, based on modality. Based on the marginal means, is there a difference between modalities
* This is independent of medication
* we found p = 0.000 = significant meaning that there is a difference between at least one modality from another modality
* partial eta squared = .896 (greater than .14) = large - most people will have a difference in their ROM depending on if they used one of those modalities - not a percentage
* Power = 1 - we are very confident in our p value (were 100% sure that we correctly identified a difference between groups)
Medication is looking to see if theres a difference between nsaids and not using naids
* This is independent of modality
* The main effect is significant
* large effect size
* Confident that there is a significant difference that will be shown in most people
Modality * Medication: This line is out itneraction effect. This is saying between these 6 groups is there at least a difference between one and another one in that group (not identifying where the difference is)
* Not significant = no special group
* small effect size
* power = low
Main effect vs Marginal means
The main effect examins the impact of one independent variable (factor) on the dependent variable, averaging over the levels of the other independet variable. For example, if you’re looking at the main effect of conservative interventions, you’re evaluating how these interventions affect the outcome w/o considering the medication interventions
Marginal means are the average values of the dependent variable for each level of an independent variable, calculated by averaging across the levels of the other factor. For instance, if you calculate the marginal mean for conservative interventions, it will reflect the average outcome for each conserative intervention, averaging over all medication internventions
Summary:
* Main effect: Focuses on the impact of one factor, ignoring the other
* Marginal means: Provides the average outcome for each level of a factor, considering all levels of the other factor
This is showing a big interaction effect between Ice and Nsaids. Meaning there was a special combination
* This is the exact same experiment ran above except showing that interaction effect
Modality line: This is the main effect of modality
* p 0.005 = significant
* So theres a significant main effect of modality. = regardless of medication theres a significant difference between the modalitites - regardless of if we use medication or no medication theres a difference between the 3 modalities = main effect of modality. Out of people who used ice, splint, or rest regardless of medication there was a signficiant difference somewhere between these modalities.
* partial eta squared = 0.181 = large = seen in most people
* power = .867 = were 86% sure that we correctly identified a difference (backs up our p value)
Medication line: This is the main effect of medication.
* P = 0.020 = significant
* So theres a signfiicant difference between using medication and not using medication regardless of modality
* partial eta swuared = 0.097 = medium = happens in some people
* Power = .65 = not great = were 65% sure that we correctly identified a difference (backs up p value)
Modality * Medication (interaction effect): This is looking to see if 1 of the 6 groups is signfiicantly different from the others (really looking to see if any 1 group is signficiantly different from any of the other groups)
* p = 0.005 = signfiicant
* So were saying that were almost 100% sure that one of the 6 combinations is significantly different from one of the other combinations.
* partial eta squared = 0.177 = large = happens in most people
* Power = 0.851 = were 85% sure that we correctly identified a difference (backs up p value)
She said that the person should do modality over medication because the main effect of modality was higher (I think she got thise from the partial eta squared)
Shes specifically concerned about the 9 circled effect
A two-way ANOVA was used to analyze the effect of medication and modality on pain free ROM. The ANOVA shows significant outcomes for the main effect modality (F(2,54) = 5.97, p = 0.005, partial eta squared = 0.181), the main effect of medication (F (1,54) = 5.77. p = 0.02, partial eta squared = 0.097), and their interaction (F(22,54) = 5.79, p = 0.005, partial eta squared = 0.177)
* Two way means 2 factors are comapred (doesnt tell me anything about levels)
* ANOVA = comparing 3 or more group means
* significant outcomes for the main effect of modality = one of the 3 modalities was significantly different from the others regardless of medication. Also has a large partial eta squared
* Significant outcomes for the main effect of medication = using medication vs no medication was significantly different regardless of modalities. moderate partial eta squared.
* Significant interaction effect, meaning there was at least one special combo that was statistically different from one other combination. Large partial eta squared
Repeated measures ANOVA = taking repeated measures within the same group
We took 1 group of people and tested their forearm strength w/ their forearm in neurtral, forearm in pronation, and forearm in supination
One-way repeated measure
* 1 factor = forearm position
* 3 levels (neutral, supination, pronation)
This is looking at the same group of 9 people for each scenario. (its repeated measures 1 way ANOVA)
Mean = the average change from baseline
NOTE:
* In a One/Two-way ANOVA were trying to compare the differences between groups (so its 3 or more groups)
Repeated measures ANOVA
* Difference in how a subject scores between different conditions
* Within subject differences - meaning its only 1 group but multiple different scenrios and seeing if theres a significant difference (must be at least 3 scenarios to utilize the ANOVA over t-test)
One/Two-way ANOVA
* Assumes homogeneity (equal) variances between groups
Repeated Measures ANOVA
* Assumes sphericity - means the exact same thing as homogeneity, just different terminiology
* Meaning, theres homogeneity of variances, but because its the same group of people, they use this term. Essentailly means between the 3 different conditions the variance of data between them is all roughly the same (they have equal variance in each level of the factor)
So if a repeated measures ANOVA is being utilized you may see “Sphericity is assumed” and that just means theres homogeneity of variance between the levels within the factor.
One/Two-way ANovas utilize Levene tests to measure the homogeneity of variance (higher p = not statsticailly different variances). However, Repeated measures ANOVA utilizes Mauchly’s Test for Sphericity to determine the variance (p = high = not stasticially different variances)
* This is done the exact same way
* you can see in the box below there sphericity p = 0.239 = not staistically different sphericity between groups = equal variance
This is the same example from above cont’d
Position line:
* P = 0.000, meaning that theres a significant difference between at least 1 of the forearm positions from another forearm position
* Partial eta squared = large, meaning most people will be impacted
* Power = 100%, meaning that were 100% sure that correctly identified a change (backing up our p value)
You can intrepret this by “forearm position affects strength so you would need to mind the position the forearm is in when testing and retesting strength [intra rater relabibility])
2 way repeated measures anova (2 factors)
* factor 1 = elbow position
* factor 2 = forearm position
also called a 3x2 repeated measures sign (named for its structure
* 3 = # of scenarios in factor 1 (3 forearm positions)
* 2 = # of scenarios in factor 2 (2 elbow positions)
* 6 total measurements being taken
Marginal mean of elbow angle of 90 degrees = 30.79. Meaning that independent of forearm position an elbow angle of 90 degrees yielided an average strength output of 30.79
Marginal mean of Pronation = 21.19. Meaning independent of elbow angle the forearm being in pronation had an average strength output of 21.19
Top line elbow angle
* this line is looking at the main effect of elbow angle independent of forearm position
* p = 0.327, meaning that theres not a significant difference between flexed and extended elbow angle
* Partial eta squared = 0.137 = moderate - meaning that elbow angle did not affect strength for most people regardless of forearm position. in the clinic this tells me that the elbow angle doesnt matter when testing strength.
* power = 0.150, meaning that were 15% sure that we correctly identified a difference between the two groups. However, our p value said their wasnt a significant difference between groups.
Forearm position
* this line is looking at the main effect of forearm position independent of elbow angle
* p = 0.013, meaning there is a significant difference between at least 1 group and another group (of forearm positions)
* partial eta squared = 0.465 = large effect size (happens in most people) - for most people forearm position will have a significant impact on force output. so make sure you’re testing and retesting in the same forearm position, howver, elbow angle doesnt seem to impact strength output
* Power = 0.804 = were 80% sure that we correctly identified a difference between the groups
Elbow x forearm position (interaction effect)
* p = .888 = not significant meaning that theres no 1 special combination that thats statsitically different from the others
* partial eta squared = small
* Power = 0.065 = were 6.5% sure that we correctly identified a difference between groups. however, we said there was no difference (p was large) so this makes sense
A two-way repeated measures ANOVA was used to look at the effect of forearm and elbow position on elbow flexor strength. The main effect of forearm position was significant (F = 6.074, p = 0.013, eta squared = 0.465), there was no difference between the two elbow positions (F = 1.112, p = 0.327, eta squared = 0.137). The interaction effect was not significant (F = 0.120, p = 0.888, eta squaraed = 0.017)
* Two way repeated measures ANOVA - means that there was 2 factors being compared (each of which had levels). And its an ANOVA so its at least comparing 3 group means.
* Main effect of forearm position was significant - meaning that independent of elbow angle there was a difference between at least 1 and another forearm position. p = 0.013 meaning this was significant. eta squared = 0.465 = strong = there was a significant difference between these groups in most people
* There was no difference between elbow positions (p = 0.327, eta squared = 0.137, meaning a moderate amount of people did not see a difference with elbow position (almost most).
* There was no interaction effect between tbe groups (p = 0.888 = not significant). eta squared = 0.017 = small, meaning only a small amount of people saw some interaction effect?
Mixed Designs ANOVA
We are now interested in:
* Between group differences
* And within group differences
So were taking repeated measures within the same group - so in this sense its repeated measures (just like we did before)
And its also a 1 way ANOVA
* I think she meant to say 2 way
* factor 1 = intervention
* factor 2 = time
24 subjects have a TBI
measuring self reported symptoms
* measure at baseline
* Measure after 6 months of intervention
* Measue 12 weeks after that
This is how lots of intervention studies are set up
baseline = marginal mean = 20.34
exercise group rehardless of time = 33.09
assuming spericity (because we have repeated measures in this)
Time line: The main effect of time regardless of intervention
* this is looking at the imapct of time regardless of intervention
* p = 0.000 = significant, meaning that there is a difference somewhere between the time groups (baseline, 6 weeks, follow-up)
* partial eta squared = large = will happen in most people
* power = 1 = we correctly identified a difference between groups
* Regardless of what intervention they did people got better over time
Time * Treatment: This is looking at the interaction effect to see if theres some magic combination of time and treatment
* p = 0.073 = not signficiant - meaning that there is no magic combo
* partial eta squared = 0.181 = large, meaning that in most people there is no interaction effect
* Power = .623, meaning that theres a 62% chance that we correctly identified a difference. However, we said there is no difference so this doesnt matter
Treatment line:
* This is looking at the main effect of treatment independent of intervention
* p = 0.097 = not signfiicant, meaning that there is not a significant difference between one of the treatment options from another
* partial eta squared = 0.199 = large meaning that in most people there will not be a significant difference between different treatment (intervention) types
* Power = 0.463 = 46% chance of correctly identifying a difference between the groups. However, we did not identify a difference between the groups so it doesnt matter
So we can conclude that independent of treatment (it doesnt really matter which one you use) as long as time goes by its going to help. - PT!!
A two-way mixed ANOVA was used to examine the cahnge in symptoms over time following TBI. The ANOVA showed a significant main effect of time (F(2,42) = 40.32, p < 0.001 n2/p = 0.658), with improvements increasing over the three time periods.
There was no significant effect of treatment (F(2,42 = 2.61, p = 0.097, n2/p = 0.199) and no significant interaction (F(4,42) = 2.32, p = 0.073, n2/p = 0.181).
* Two-way mixed ANOVA - meaning that the study design is looking at some kind repeated measure and some other factor (in our case it was time [which was essentially the factor w/ repeated measures] and intervention)
* ANOVA = 3 more means comapred
* change in symptoms over time tells you that our repeated measure was time. Were doing it over time. - thats the factor here
* ANOVA showed a signfiicant main effect of time, meaning that there was a difference between at least 1 of the time groups and another time group on overall TBI baring interventions
* There was no significant effect of treatment baring time, meaning there main effect of interventions was negliabile (1 wasnt signfiicantly different from another)
* There was no significant interaction (no magic combo)
This table is how the above example would be reported in an actaul artical. This isnt how its going to be done on an exam.
