Lecture 5

Question 1

What are the null and research hypotheses for ANOVA in notation?

Answer 1

H0: μ1 = μ2 = μ3
H1: μ1 =/= μ2 =/= μ3

Question 2

What are the null and research hypotheses for ANOVA in words?

Answer 2

H0: No difference between the means.
H1: There is at least one mean difference in all of these means/between the groups.

Question 3

Which hypothesis do we test?

Answer 3

We always test the null, always. No exceptions.

Question 4

How do we make the decision to reject or fail to reject H0?

Answer 4

Distribution of F statistic under H0. For continuous variables, area under the curve = 1. Because we turned it into a percent, then into probability.
Fcritical separates rejection region from the rest of the curve. If F falls in the rejection region, then we reject the null.
Rejection region = alpha = .05 (unless told otherwise).
If F is less than .05, we reject H0. Obtained value, Fobtained, or Fobserved, comes from our data.
Fobtained in summary table (from data), Fcrit comes from F table.
Compare Fobtained to Fcrit, if Fobtained is greater than Fcrit we reject H0.

Question 5

What do we say if Fobtained is less than .05?

Answer 5

We say that the difference between means is statistically significant.

Question 6

How is significance graded?

Answer 6

There is no gradation of significance, it is only either above or below Fcrit (i.e. either significant or not significant).

Question 7

When writing a conclusion about rejecting H0, what language must be included?

Answer 7

Words such as “probably”, “it is likely”, or “it seems that”.
This is because the area under the rejection region is a probability, so will be wrong 5% of the time. So we will be rejecting H0 when it’s actually true 5% of the time.

Question 8

The conclusions from any stat test may be in error. True or false

Answer 8

True, we’ll never know for sure if the results from any stat test are in error or not, but it is always possible that they are in error.

Question 9

What are the two possible errors that could be made when making a conclusion from a statistical test?

Answer 9

Type I or Type II error.

Question 10

What is a type I error?

Answer 10

• Alpha
• By convention .05
  Means when rejecting H0 when H0 is true.
  If alpha = .05, F5 exceeding Fcrit will occur 5% of the time due to chance alone.
  So 5% of the time when we accept H1, we’ll be wrong.

Question 11

What is Type II error?

Answer 11

• Beta
• “accepting” H0, when H0 is false.
• (We actually don’t ever “accept” it, we fail to reject it)
• We can never know the value of Beta for sure, because we need to know parameters to determine it.

Question 12

Describe a 2X2 contiingency table

Answer 12

Truth: either H0 is true, or H0 is false
Top:
H0 true | H0 false

Sides:
“Accept” H0 decision
|
Reject H0

For box H0 true + “accept” H0 decision: 1 - alpha
For box H0 false + “accept” H0 decision: Type II error - Beta
H0 true + reject H0: Type I error - alpha
H0 false + reject H0: 1 - Beta

Question 13

What is power?

Answer 13

Correct rejection off H0

Acceptable power values range from 0.7 to 0.8

Question 14

Give a close up of power, why is having a high power important?

Answer 14

If spending lots of time and money on a study, want a 70-80 percent chance of finding a significant difference if one exists.
- need a measure of effect size.

Question 15

Recall Cohen’s d

Answer 15

d = population effect size
d = (μ1 - μ2)/σ
dcarrot = (Xbar1 - Xbar 2)/σcarrot
Therefore d = distance between means in standard deviation units.
so dcarrot = .4 means that group means differed by .4 of a standard deviation.

Question 16

What does power equal?

Answer 16

Power = degree of non overlap, how many standard deviations groups differ.
d = .2 = small
d = .5 = medium
d = .8 = large
d = 1.5 = very large
d = 2.2 = huge (international superstar level psychologist)

Question 17

What is squiggly fcarrot?

Answer 17

effect size for ANOVA

Question 18

squiggly fcarrot =

Answer 18

√(K-1)F/N or √(a-1)F/N where K and a stand for number of groups.

Question 19

squiggly fcarrot ranges

Answer 19

~.1 = small
~ .25 = medium
~ .4 = large

Question 20

What are the two ways that we use power?

Answer 20

Post hoc (after the fact) and apriori.

Question 21

Describe post hoc determinations of power

Answer 21

• use effect size, n, and power table to calculate power.
• eg:
  n1 = 15
  n2 = 13
  n 3 = 20
  n4 = 17
  F = 1.4
  N = 65
  alpha .05
  average n = 16.25 –> 16
  U = K-1 = 3

squiggly fcarrot = √(K-1)F/N = √(4-1)1.4/65 = √(3)0.2153 = √0.064615384 = 0.254 (medium) - this is the effect size.
Power: in table C.3 = .34!
The power was inadequate to detect a medium effect size - need to replicate study with a more adequate N!!

Question 22

Describe apriori power tests:

Answer 22

• Tells us how many participants per group are needed to have adequate power.
• Slight problem: need an expected effect size to determine n for a specified power at some alpha level.

Question 23

How do we overcome the problem of apriori power tests?

Answer 23

• Past research –> i.e. average of squiggly fcarrot’s from similar studies –> similar treatment, treatment duration, type of subjects, instruments used to measure DV, etc. Better to have some estimate than no estimate.

Question 24

How do you find an apriori estimation of power?

Answer 24

estimated squiggly fcarrot = .25
power = .07
alpha = .05
u = 3
table C.3 - we need 36 participants per group are needed.

Question 25

What are four ways to improve power?

Answer 25

1. Adopt a more lenient alpha (e.g. alpha = .10)
2. Use a one tailed test when literature supports a directional hypothesis
3. Decrease within group variability
   - sample selection (homogeneous group)
   - repeated measures designs
4. Have equal n

Question 26

What is the harmonic mean? When is it used?

Answer 26

With unequal n, rather than use the arithmatic mean of the sample sizes, we use the harmonic mean
- General form of the equation:
Harmonic mean = # of treatment groups/Σ(1/sample size)

So for two samples:
nh = 2/1/n1+1/n2