ANOVA Analysis of Variance

 Basics of parametric statistics  ANOVA – Analysis of Variance  T-Test and ANOVA in SPSS  Lunch  T-test in SPSS  ANOVA in SPSS

 The arithmetic mean can only be derived from interval or ratio measurements.  Interval data – equal intervals on a scale; intervals between different points on a scale represent the difference between all points on the scale.  Ratio Data – has the same property as interval data, however the ratios must make mutually sense. Example 40 degrees is not twice as hot as 20 degrees; reason the celsius scale does not have an absolute zero.

 Assumption 1: Homogeneity of variance – means should be equally accurate.  Assumption 2: In repeated measure designs: Sphericity assumption.  Assumption 3: Normal Distribution

 Assumption 1:Homogeneity of variance  The spread of scores in each sample should be roughly similar  Tested using Levene´s test  Assumption 2:The sphericity assumption  Tested using Mauchly´s test  Basically the same thing: homogeneity of variance

 Assumption 3: Normal Distribution.  In SPSS this can be checked by using: ▪ Kolmogorov-Smirnov test ▪ Shapiro-Wilkes test  These compare a sample set of scores to a normally distributed set of scores with the same mean and standard deviation.  If (p> 0.05) The distribution is not significantly different from a normal distribution  If (p< 0.05) The distribution is significantly different from a normal distribution

 Difference between t-test and ANOVA:  t-test is used to analyze the difference between TWO levels of an independent variable.  ANOVA is used to analyze the difference between MULTIPLE levels of an independent variable.

 Independent variable = apple  Dependent variables could be: sweetness, decay time etc. t-test ANOVA …or more

 The ANOVA tests for an overall effect, not the specific differences between groups.  To find the specific differences use either planned comparisons or post hoc test.  Planned comparisons are used when a preceding assumptions about the results exists.  Post Hoc analysis is done subsequent to data collection and inspection.

 A Post Hoc Analysis is somewhat the same as doing a lot of t-tests with a low significance cut- of point, the Type I error is controlled at 5%.  Type I error: Fisher’s criterion states that there is a o.o5 probability that any significance is due to diversity in samples rather than the experimental manipulation – the α-level.  Using a Bonferroni correction adjusts the α-level according to number of tests done (2 test = o.5/2 = test= 0.5/5= 0.01). Basically the more tests you do the lower the cut of point.

 Variation in a set of scores comes from two sources:  Random variation from the subjects themselves (due to individual variations in motivation, aptitude, etc.)  Systematic variation produced by the experimental manipulation.

 ANOVA compares the amount of systematic variation to the amount of random variation, to produce an F-ratio: systematic variation random variation (‘error’) F =

 Large value of F: a lot of the overall variation in scores is due to the experimental manipulation, rather than to random variation between subjects.  Small value of F: the variation in scores produced by the experimental manipulation is small, compared to random variation between subjects.

 In practice, ANOVA is based on the variance of the scores. The variance is the standard deviation squared: variance

 We want to take into account the number of subjects and number of groups. Therefore, we use only the top line of the variance formula (the "Sum of Squares", or "SS"):  We divide this by the appropriate "degrees of freedom" (usually the number of groups or subjects minus 1). sum of squares

 Between groups SSM: a measure of the amount of variation between the groups. (This is due to our experimental manipulation).

 Within GroupsR: a measure of the amount of variation within the groups. ( This cannot be due to our experimental manipulation, because we did the same thing to everyone within each group ).

 Total sum of squares:a measure of the total amount of variation amongst all the scores. (Total SS) = (Between-groups SS) + (Within-groups SS)

 The bigger the F-ratio, the less likely it is to have arisen merely by chance.  Use the between-groups and within-groups d.f. to find the critical value of F.  Your F is significant if it is equal to or larger than the critical value in the table.

Here, look up the critical F- value for 3 and 16 d.f. Columns correspond to between-groups d.f.; rows correspond to within-groups d.f. Here, go along 3 and down 16: critical F is at the intersection. Our obtained F, 25.13, is bigger than 3.24; it is therefore significant at p<.05. (Actually it’s bigger than 9.01, the critical value for a p of 0.001).

 One –Way ANOVA  Independent  Repeated Measures  Two-way ANOVA  Independent  Mixed  Repeated Measures  N-way ANOVA

 One-Way: ONE INDEPENDENT VARIABLE  Independent: 1 participant = 1 piece of data. Independent variable: Yoga Pose,3 levels Dependent variables: Heart rate, oxygen saturation

 One-Way: ONE INDEPENDENT VARIABLE  Dependent : 1 participant = Multiple pieces of data. Independent variable:Cake,3 levels Dependent variables: Blood sugar, pH-balance

 Two-Way : TWO INDEPENDENT VARIABLES  Independent : 1 participant = 1 piece of data. Independent variables: Age, Music Style >40<40 Indie-Rock Classic Pop

 Two-Way : TWO INDEPENDENT VARIABLES  Mixed:  Variable 1: Independent (Controller)  Variable 2: Repeated measures (Space Ship)

 Two-Way: TWO INDEPENDENT VARIABLES  Dependent : 1 participant = Multiple pieces of data. Independent variables: Exercise, Temperature 20 °25 ° 30 °

Running SPSS (repeated measures t-test)

Interpreting SPSS output (repeated measures t-test)

RUNNING SPSS

Click ‘Options…’ Then Click Boxes: Descriptive; Homogeneity of variance test; Means plot

SPSS output

 One-way independent-measures ANOVA enables comparisons between 3 or more groups that represent different levels of one independent variable.  A parametric test, so the data must be interval or ratio scores; be normally distributed; and show homogeneity of variance.  ANOVA avoids increasing the risk of a Type 1 error.