W8- Introduction to analysis of variance (ANOVA)

used when there are 3 categories)

(ANOVA -

· ANOVA and t‐test are similar - just better with ANOVA - Compare means between‐groups · With 2 groups both work but: üt‐test more efficient ûANOVA inefficient · With more than 2 groups: ût‐test not efficient üANOVA more efficient

ANOVA and t‐test

different groupings of the IV · These are the independent variable(s) In our example, the factor ismusic type

ANOVA terminology Factors -

Between‐groups variance is the variation (difference) between mean scores in each condition There is a difference in test scores between the groups à9 to 22.2 · When the means are very different, we say that there is a greater degree of variation between the conditions · If there were no differences between the means, there would be no variation

Between‐groups variance

· Between‐groups variance arises from these sources: · If means were all the same = we would say no differences

Between‐groups variance

· This is the effect of the IV(s)- what we are actually trying to measure · We want a difference between experimental conditions - the scores of participants in one group are different to scores of participants in another group · We are measuring whether the variable we are looking at is actually having an effect

Between‐groups variance Treatment effects

· These are factors that vary between participants. So in our example, each participant will only experience one level of a factor - either no noise, intermittent noise or constant noise. This is a between‐subjects design (like an independent sample t‐test)

Between‐subjects factors

Put another way we can represent this as:F = variance due to manipulation of IV error variance If the error variance is small compared to variance due to manipulation of the IV, the F‐ratio will be greater than 1 On the other hand, if the effect of the IV is small and/or the error variance is large, the F‐ratio will be less than 1• An F ratio of less than 1 indicates that the effect of the IV is not significant The greater the F‐ratio the better

Calculating the F-ratio

· We want to see if our manipulation of the IV is responsible for the differences between scores (rather than being due to error variance) · To do this we need to calculate the ratio of the variance due to our manipulation of the IV (between‐groups variance) and the error variance (within‐groups variance) · This ratio is called the F‐ratio ('F' stands for a statistician named Fisher)

Calculating the F‐ratio (came up by fisher)

We need to learn some new terminology in order to fully describe the different kinds of ANOVA designs

Describing ANOVA designs

If the F‐ratio is larger than 1, we need to decide if the value is large enough to be statistically significant SPSS can calculate whether the effect of the IV is sufficiently larger than the effect of the errors SPSS will report the exact p‐level for a given F‐ratio It takes in to account the number of observations (degrees of freedom) The p value is the probability of getting this F ratio by chance alone. The p value needs to be equal to or less than 0.05 for the F ratio to be regarded as significant

How do we find out if the F‐ratio is significant?

Variability between everything, but also looks at in group differences Analysis of Variance analyses the different sources from which variations in scores arise It looks at the variability between conditions (between‐ groups variance) and within conditions (within‐group variance) It tries to determine whether we have a true effect of the IV rather than an effect of individual differences (variance between conditions greater than variance within conditions)

How does ANOVA work?

When describing an ANOVA design we need to specify: 1. How many factors are involved in the design 2. How many levels there are in each factor 3. Whether the factor(s) are within or between subjects

How to describe ANOVA designs

Even if they are being treated the same = they will respond differently and cause differences People within the same group differ even though they are treated the same It is highly likely to have some level of within‐groups variance because people naturally differ e.g., knowledge, IQ, personality etc. E.g., asking people to remember a list of words. Group 1 is asked to use a specific strategy. Group 2 is not told to use a specific strategy. Therefore the participants in Group 2 might choose to use different strategies to remember the list causing increased variation in performance

Individual differences

People naturally vary but we don't want a high amount of individual differences as this might lead us to think our IV is having an effect when it is actually due to differences between the ability of participants in different groups E.g., we want to investigate reaction time to identify famous paintings. Group 1 see historic paintings and Group 2 see modern paintings. However, Group 2 contains several students taking a Modern Art degree We find performance in Group 2 is quicker so we attribute this to it being easier to identity modern paintings when it is probably due to differences between the group skill levels

Individual differences

· These are like conditions. In our example we have three levels of the factor: constant low level of music, no music and intermittent music

Levels of factors

· One or more between and within subject factors · The term mixed ANOVA issued when a study design includes one or more within subjects factors and one or more between subjects factors. So perhaps we want to look at male and female test scores (between‐subjects) for each of the facial expressions (within‐subjects)

Mixed ANOVA designs

· Can happen through errors of measurement - it can be illuminated - E.g everyone come in to do a treadmill test at the same time : this way it will reduce error · Errors of measurement can arise from a variety of sources such as: · Varying external conditions - differences in time of day at testing · State of the participant - current focus of attention, motivation · Experimenter's, or computer's, ability to measure and score accurately

Random errors

· Errors of measurement can arise from a variety of sources such as: - Varying external conditions - differences in time of day at testing - State of the participant - current focus of attention, motivation - Experimenter's, or computer's, ability to measure and score accurately

Random errors

Analysis of Variance (ANOVA) is a parametric equivalent to t‐tests that involves more than two groups There are other versions Both the t‐test and ANOVA compare means but the t‐test is limited to two groups as the risk of making the Type 1 error increases when dealing with more than two groups ANOVA has number of assumptions, namely; the data is interval or ratio, the data is normally distributed, the scores have equal variances (homogeneity), and for independent‐samples the participants must be randomly sampled ANOVA calculates the variances for both between‐group and within‐group to see if differences are due to the treatment condition and computes F‐ratio There is specific terminology in ANOVA that helps to identify the type of design used

Summary

· According to the logic of ANOVA - subjects in different groups should have different scores because they have been treated differently (i.e. given different experimental conditions) but subjects within the same group should have the same score - all down to intervention you have exposed them to · We want to know if our different treatment groups have different scores because of the treatment condition they have been exposed to • I.e., we are trying to find out if the variance or spread of scores is larger between the groups (i.e. in the different conditions) than the spread within the groups If the variance between conditions is much larger than the variance within conditions we can say that our IV is having a larger effect on scores than the individual differences are This comparison of variance due to nuisance factors (error variance, individual differences) compared to variance due to our experimental manipulation is called partitioning the variance

The Logic of ANOVA

1. How many factors are involved in the design · The number indicates the number of factors (IVs) - One‐way ANOVA (one factor, e.g. facial expression) - Two‐way ANOVA (two factors, e.g. facial expression and gender) - Three‐way ANOVA (three factors, e.g. facial expression, gender and age group) - and so on for 4, 5, 6... etc. factors 2. How many levels there are in each factor · We could describe our 2-way ANOVA example (two factors, e.g. gender and facial expression) asa 2 x 4 ANOVA - Gender has 2 levels (male or female) - Facial expression has 4 levels (neutral, angry, happy, and sad)

Types of Analysis of Variance (ANOVA)

1. The dependent variable (DV) consists of data measured at interval or ratio level 2. The data are drawn from a population which is normally distributed 3. There is homogeneity of variance (the samples being compared are drawn from populations with the same variance) 4. For independent groups designs, independent random samples must have been taken from each population

What are the assumptions for ANOVA?

1. The dependent variable (DV) consists of data measured at interval or ratio level Interval: difference between points but no true zero e.g. temp Ratio: e.g. test score - can have a zero - meaningful difference between numbers

What are the assumptions for ANOVA?

ANOVA is a parametric test used to test for differences Used more than 1 independent variable Allows you to look at multi in time in combination = see if it's a true independent variable or whether its different factors. Used when we have more than two groups and / or more than one independent variable (factor) A major advantage of ANOVA is that it allows you to investigate the effect of multiple factors on your dependent variable at the same time (in combination) It essentially tries to determine whether we have a true effect of the IV rather than an effect of individual difference (variance between conditions greater than variance within conditions)

What is ANOVA and why do we need it?

Every time you conduct a t‐test using a .05 level of significance, there is a 5% probability of falsely rejecting the null hypothesis (a Type 1 error) By running multiple tests you increase the chance of making a Type 1 error: p = 1 - (0.95)n where n = no. of comparisons made E.g. making 3 comparisons = 1 - (0.95)3 = 1 - 0.857 = 14.3% This is called the experiment wise error rate (also known as familywise error rate) An ANOVA controls for these errors so that the Type 1 error remains at 5% and you can be more confident that any significant result you find is not just down to chance

Why not use several t-tests?

Every time you do t-test you have error - you are inflating your chances of getting an error = familywise error rate · Imagine that we have a study with 3 experimental conditions and we are interested in differences between these three groups · If we just used t‐tests on each pair of groups we would carry out three separate tests: • Group1vsGroup2 • Group1vsGroup3 • Group2vsGroup3

Why not use several t‐tests?

· Within‐groups variance can be called error variance · The variability within the groups is not produced by the experiment and that is why it is considered error variance · It can arise from these sources:

Within-groups variance

Condition 1 has no variation within it (all participants have scored the same) Condition 2 has little variation (from 15 to 16) Condition 3 has greatest variation (from 17 to 26

Within‐groups variance

Variational difference between people in the same group Within‐groups variance is the variation (difference) between people within the same group There is a difference in test scores within the groups

Within‐groups variance

· Might vary between particiaptns - ages, genders · These are factors that vary within a participant. In a different study, we may wish to administer all of the noise conditions to each participant to see how they perform. This is a within‐subjects design (like a repeated measure t‐test) · Repeated measures type design

Within‐subjects' factors

2. The data are drawn from a population which is normally distributed

assumption 2

3. There is homogeneity of variance (the samples being compared are drawn from populations with the same variance)

assumption 3

4.For independent groups designs, independent random samples must have been taken from each population

assumption 4

scale where differences between points are equal but no true zero e.g. temp

interval

catergorical varibale so values cannot be ranked e.g. eye colour

nominal

qualitive variable, its values can be ranked but differences between points are not equal e.g. likert scale

ordinal

interval scale but with true zero e.g. memory test

ratio