ONE WAY INDEPENDENT GROUPS ANOVA

ONE WAY INDEPENDENT GROUPS ANOVA

A one-way ANOVA is focused on the effect of one IV on the DV, but the IV can have more than two levels 

ADVANTAGE OF ONEWAY INDEPENDENT ANOVA

• Guards against familywise error  

  • Analyses all the variance in data at once 

  • Using multiple t-tests inflates type 1 error rate 

• Is an omnibus technique  

  • Tests whether DV varies among the levels of the IV 

  • Tells us whether there is a significant difference between group means somewhere 

  • Does not tell us which means are significantly different 

IF ANOVA YEILDS SIGNIFICANT RESPONSE

We do follow up tests to find what means are significantly different  

IF ANOVA DOES NOT YEILD SIGNIFICANT RESPONSE

We state there is no effect of IV on DV, no follow up tests.  

ONE WAY INDEPENDENT ANOVA FACTORS

IV  

• Called the factor of treatment (if manipulated) 

LEVELS 

• Different conditions that make up a factor 

ONE WAY HYPOTHESES

NULL HYPOTHESIS  

H₀ = u₁ = u₂ = u_k 

(if three means then.. u₁ = u₂ = u₃) 

ALTERNATIVE HYPOTHESIS  

H₁ = u_k ≠ u_k' 

(at least two means are different) 

LOGIC OF ANOVA

• Observed differences relative to expected differences  

• Separate total variance into two components: 

  • Between-groups variance  

  • Within-groups variance  

BETWEEN GROUPS VARIABILITY DUE TO:

• Treatment effect/levels of factor 

• Individual differences  

• Experimental error  

WITHIN GROUPS VARIABILITY DUE TO:

• Individual differences  

• Experimental error  

TREATMENT EFFECT

• Therefore everything will cancel out apart from treatment effect  

• So if between groups variability > within-groups variability = presence of treatment effect 

MEAN SQUARES

Within-group variability: 

• MS_error, also referred to as   MS _residual 

Between-groups variability: 

• MS_treatment, also referred to as MS_model  

COMPARING MEAN SQUARES 

• If H₀ is false, there will be more variation among the means that can be accounted for by chance, and MS_treatment will be larger than MS_error 

F DISTRIBUTION

• The F-statistic aims to compare the variance among the treatments, to the variance within the samples themselves. 

• F_obt is considered significant if F_obt > F_crit 

• F distribution dependent on df  

• F_obt can only be positive  

• F distribution is positively skewed  

  • Most Fs cluster around 1 (H₀) 

• F-test is a one tailed test 

  • We are only testing for the presence/absence of treatment effect 

ONE WAY INDEPENDENT GROUPS STEPS

1. State H0 and H1 in words and symbols

2. Calculate sum of squares (TOTAL, TREATMENT, ERROR)

3. Calculate degrees of freedom (TREATMENT, ERROR)

4. Calculate mean squares

5. Calculate F ratio

6. Construct summary table

7. Find T crit

8. Make decision

9. Interpret results

EFFECT SIZE - OMEGA SQUARED

• An estimate of the proportion of variance in the population that can be accounted for by the IV 

• Small effect = .01 

• Medium effect = .06 

• Large effect = .15 

MEANING OF ONE-WAY INDEPENDENT GROUPS

ONE WAY

• single independent variable / factor

INDEPENDENT GROUPS

• aka. between subjects

• different participants in each condition

ASSUMPTIONS

1. Normality - each group from normally distributed population

2. Homeogeneity - populations have equal variances

3. Independence - each participants score is independent of each others

WHY USE INSTEAD OF T-TESTS?

If you compare three groups using t-tests you have a 5% chance of Type 1 error PER TEST.

ANOVA runs a single test across all groups simultaneously, keeping Type 1 error controlled at a = .05.

PARTIONING VARIANCE

SS TOTAL = SS TREATMENT + SS ERROR

DF FOR F CRITICAL

Numerator = df TREATMENT

Denominator = df ERROR

*Use lower df in table to be conservative

*both df’s go in brackets when reporting Fobt/crit

STATISTICAL MODEL

RELATIONSHIP BETWEEN T AND F

F_crit = t_crit² 

F = t² 

ETA SQUARED (BETWEEN SUBJECTS)

η² = SS treatment / SS total