ANOVA derivations

Question 1

How do we use degrees of freedom?

Answer 1

The degrees of freedom (df) associated with a sum of squares (SS) represent the number of scores with independent information which enter into the calculation of the SS

Question 2

what are degrees of freedom?

Answer 2

The df is the number of distinct “quantities” that are used to describe your data (sample size) subtracted by all the “constraints” that the data must satisfy.

Question 3

how do we calculate the f-ratio?

Answer 3

variance between group/Variance within group

Question 4

what dies the F ratio allow us to estimate?

Answer 4

the treatment effect after accounting for the error

Question 5

are there treatment effects if H0 is true?

Answer 5

there is no treatment effect

Question 6

are there treatment effects if H0 is false?

Answer 6

• there is a treatment effect
  treatment effect + experimental error / experimental error > 1

Question 7

what should happen to the F ratio if H0 is false?

Answer 7

the F-ratio should be large; a large ratio is unlikely if the real treatment effect does not exist (i.e., in the population)
- To test this, we compare it to the F-distribution

Question 8

what is the F distribution?

Answer 8

an estimate of how likely it is to obtain an F-ratio given that 𝐻0 is true, with X groups of size 𝑁 (these determine the degrees of freedom)

Question 9

when do you reject H0 using the F ratio?

Answer 9

• when the F oberverd value > F critical value
• otherwise do not reject

Question 10

how do you calculate the critical F value?

Answer 10

• An F table, for a threshold α (e.g., 𝛼=0.05) contains critical F-values for specific degrees of freedom
• Effect degrees of freedom (〖𝑑𝑓〗_𝐴) are listed in columns
  Error degrees of freedom (〖𝑑𝑓〗_𝑅) are listed in rows

Question 11

what do you assume when conducting an ANOVA?

Answer 11

• Independence
• Treatment levels are randomly assigned to subjects; i.e., there is no systematic difference in how they are assigned (recall the lab temperature example from Lecture 1)
• Normality
• The DV is normally distributed in the population
• Homogeneity of variance
• All the treatment populations have the same variance (i.e., only the means differ)

Question 12

what does the assumption of independence rely on?

Answer 12

the research design - random sampling (assignment of subjects to treatments) makes it probable that samples are independent

Question 13

what is a common test for homogeneity of variance in between-group variance design?

Answer 13

• Barlett test

Question 14

what is a common test for homogeneity in mixed or within-group design?

Answer 14

Box’s M

Question 15

what are the common tests for normality?

Answer 15

• Skewness
• Kolmogorov-Smirnov
• Shapiro-Wilk
• These tests compare the sample data distribution to a theoretical normal distribution
  *They are very sensitive with large samples

Question 16

what is skewness?

Answer 16

Skewness means values are more probable on one side of the mean than the other. A perfectly normal distribution has a skewness of zero.

Question 17

what direction does a negative skewness go in?

Answer 17

leftward (skweness <0), peak to right

Question 18

what direction does a positive skewness go in?

Answer 18

rightward (skewness >0), peak to the left

Question 19

what is a Z-test?

Answer 19

tests whether the skewness is significantly different from zero.

Question 20

how is Z calculated?

Answer 20

𝑧=𝛾/𝑆𝐸_𝛾
where y = skewness value
SE = standard error

Question 21

how is standard error calculated?

Answer 21

𝑆𝐸𝛾=√(6⁄𝑁)
- where N is the sample size

Question 22

what is the null hypothesis when testing the skewness of a distribution?

Answer 22

• states that the population distribution is not skewed (𝛾=0 and therefore 𝑧=0)
• If we find that |𝑧|>1.96*, we conclude that the population is skewed; and therefore not normally distributed

Question 23

what is a monotopic transformation of data?

Answer 23

• preserve the order of our data points
• required when transforming data in order to make it normally distributed or to make variance homogeneous

Question 24

what does rata transformation do to the likely hood of type I and type II errors?

Answer 24

• reduces the chance of making type II errors (false negative)
• failing to transform data will not increase the chance of a Type I error (false positive)

Question 25

when should data be transformed?

Answer 25

Data should be transformed whenever there is a substantial skew or heterogeneity of variance