Week 5-Simple ANOVA Flashcards
What is a One-way ANOVA?
-Previous lectures have looked at predicting variance (the relationship) in continuous, normally distributed dependent variables, as well as dichotomous variables (just a fancy way of saying linear regressions).
-ANOVA is an extension of this – we are using ANALYSIS OF VARIANCE to tell us something about means.
-ANOVA is basically the same as regression – they are both LINEAR MODELS. If you understand one you understand the other
‘T-Tests compare means, let’s just do a couple of them instead…’ What is the problem with this statement?
Doing multiple t-tests would cause an inflated type 1 error rate:
-We inflate the chances of finding a significant p-value (a p-value less than 0.05).
-P-value is the probability of observing results as extreme (or more) as observed, if the null hypothesis is true.
-We use the criteria p < .05 in psychology.
-Running multiple t-tests would mean that we are getting multiple attempts to find a significant p-value (just throwing the p-value out of the window)
-The more attempts, the greater the likelihood of obtaining a p-value less than .05.
-This completely undermines the p <.05 criteria.
How does the exact same principle applies to p-values?
Familywise error rate. FWE = 1 – (1 – α)NUMBER_OF_TESTS:
-Type 1 Error rate (falsely rejecting H0)
- alpha (or α = .05) is for 1 test.
An example… 3 groups will need 3 t-tests (A vs B; A vs C; B vs C):
(i) 0.953 = 0.857
(ii) 1 – 0.857 = 0.143
Now 5 groups? (A vs B; A vs C; A vs D; A vs E; B vs C; B vs D…):
(i) 0.9510 = 0.598
(ii) 1 – 0.598 = 0.402
-FWE=the probability of rejecting the null hypothesis falsely aka a false positive
-a=alpha level x number of tests you need to run
What are Type 1 and Type 2 errors put simply?
-Type I error (false positive)=says there are results when there aren’t
-Type II error (false negative)=says there aren’t results when there are
How does ANOVA work?
-ANOVA protects the familywise error rate
-ANOVA is an OMNIBUS test
-It tells us if there a significant effect, but does not tell us where the effect is (i.e. is there a difference between group A and B, group B and C, group A and C… etc.
-If our ANOVA is not significant we are not justified in examining group contrasts individually.
-It does not tell where the differences are
-You don’t look further if insignificant as ANOVA is telling you there is nothing there
What is a recap of the ANOVA Model?
-SSM = total variation that the model explains (between group variation)
-SSE = total variation due to unmeasured factors (within-group variation)
-However, these measures use a different number of observations to calculate their totals
-With 3 group, SSM compares the difference between each group mean with the grand mean (3 values)
-SSE compares the difference between each individual data point with the group mean. As there will be more than 1 participant per group, the number of values used to calculate SSE will be greater than when calculating SSM.
-This difference in values may produce a bias.
-We therefore need to correct for this.
-Not necessary to know how to run an ANOVA
How do we correct for the sum of squares values?
-We correct by dividing the SSM and SSE by their degrees of freedom.
-Degrees of freedom for SSM = k -1 (k = number of groups)
-Degrees of freedom for SSE = N – k (N = total sample size)
-Let’s say we have an N of 15 and 3 groups
-Degrees of freedom for SSM = k-1 = 3 -1 = 2
-Degrees of freedom for SSE = N – K = 15 – 3 = 12
-With this information, we can adjust our SSM and SSE values appropriately.
What are the Assumptions of one-way ANOVA?
-DV should be at the scale level
-Data should be normally distributed
-Equal variances between groups
-Scale=interval or ratio data
-Equal variance=homogenity of variance
-GROUPS must be INDEPENDENT
-Scores at one level do not influence scores at another level
What is the Hypotheses of a typical ANOVA?
-H0: There is no significant difference between mean 1, mean 2 and mean 3 (or more if you have them)
-H1: There is a significant difference between the means.
-H0=null hypothesis
-H1=alternative hypothesis
-If we have a specific direction it could be ‘scores for mean 1 are greater than scores for mean 2’.
Whether the hypothesis is directional or not is important for the test you perform!
What are the Planned Comparison Assumptions?
-Independence – contrasts should not interfere with each other
-Only ever make two comparisons at once. Why?
-Comparisons = K – 1.
-K = number of conditions.
-Not hard to understand but SPSS makes it confusing
-Telling it advance what comparisons we want to make
-Should have 2 comparisons
-If you make more than 2 comparisons, you’re just at the start of the ANOVA again
What are Planned comparison assumptions? Directional hypothesis
-Independence – contrasts should not interfere with each other
-Comparisons = K – 1.
-K = number of conditions.
-Should have 2 comparisons
-If you make more than 2 comparisons, you’re just at the start of the ANOVA again
How do we report the results of a simple one-way ANOVA? non-directional and directional
Non-directional:
‘A one-way ANOVA was conducted to compare the effect of [IV: in this case lecture type] on [DV: in this case subjective well-being]. There was a statistically significant effect of lecture type on subjective wellbeing (F(2,87) = 16.77, p < .001, ηp2 = .278). Bonferroni adjusted post-hoc tests demonstrated a significant difference between Statistics Lectures and Zoology (p = .006), Statistics lectures and Sports Science (p < .001), and Sports Science and Zoology (p = .032).
Directional:
‘A one-way ANOVA was conducted to compare the effect of [IV: in this case lecture type] on [DV: in this case subjective well-being]. There was a statistically significant effect of lecture type on subjective wellbeing (F(2,87) = 16.77, p < .001, ηp2 = .278). Planned contrasts demonstrated a significant difference between Statistics Lectures and Zoology and Sports Science (t(87) = 5.17, p < .001, d = 2.31). There was also a significant difference between Zoology and Sports Science (t(87) = 2.61, p = .011, d = 0.67).
-REMEMBER, WE ALSO HAVE TO PROVIDE INTERPRETATION OF THE DIFFERENCES! PROVIDE MEANS/SDS IN A TABLE OR A FIGURE, OR IN THE TEXT.
-Because ours was significant in non-directional, we have to do further tests
What was an In text example of a one-way ANOVA?
‘A one-way ANOVA was conducted to compare the effect of [IV: in this case lecture type] on [DV: in this case subjective well-being]. There was a statistically significant effect of lecture type on subjective wellbeing (F(2,87) = 16.77, p < .001, ηp2 = .278). Bonferroni adjusted post-hoc tests demonstrated a significant difference between Statistics Lectures (M = 62.63, SD = 16.90) and Zoology (M = 49.23, SD = 13.74; p = .006), Statistics lectures and Sports Science (M = 38.23, SD = 18.08; p < .001), and Sports Science and Zoology (p = .032).
-Same ANOVA as slide before except we just need to add mean and standard deviation as shown above
What are things to consider when ‘grouping’ participants?
-Individual differences
-Our data set demonstrates that subjective wellbeing is greater following a Statistics Lecture compared to a Zoology or Sports Science lecture.
-However, could this be confounded by a third variable. We measured familiarity with the subject… it could be that individuals who were already familiar with statistics (through A-Level perhaps) find it calming or really understand it.
What is a Summary of a One-way ANOVA
One-way ANOVA is a useful tool when you want to compare the means of more than two groups, without increasing the likelihood of TYPE 1 ERROR
Data should be continuous and normally distributed.
Alone, one-way ANOVA tells us only that an effect exists. We have to use planned contrasts or post-hoc tests to identify where the difference(s) is / are.
Provides us with an effect size (partial eta squared), which tells us how much variance can be explained by our groups. This is very similar to R squared in regression.
