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Where are we?. What we have covered: - How to write a primary research paper. What we have covered: - How to write a primary research paper - How to keep a research notebook. What we have covered: - How to write a primary research paper - How to keep a research notebook

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slide3

What we have covered:

- How to write a primary research paper

slide4

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

slide5

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

- Types of variables and scales

slide6

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

- Types of variables and scales

- Types of distributions

slide7

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

- Types of variables and scales

- Types of distributions

- Attributes of the normal distribution

- testing single values – “z” test

(one sample t-test)

slide8

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

- Types of variables and scales

- Types of distributions

- Attributes of the normal distribution

- Comparing means of two independent groups

- t-test and MWU

slide9

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

- Types of variables and scales

- Types of distributions

- Attributes of the normal distribution

- Comparing means of two independent groups

- Experimental designs

slide10

What we have covered:

- How to write a primary research paper

- How to keep a research notebook

- Types of variables and scales

- Types of distributions

- Attributes of the normal distribution

- Comparing means of two independent groups

- Experimental designs

- Comparing > 2 groups, and multiple effects

- ANOVA

slide12

Differences Between Means

  • A. 1 Sample
  • one sample t-test (z test)
  • B. 2 samples
  • 1. Independent Groups
  • - t-test (parametric)
  • - MWU (non-parametric)
  • 2. Related Samples
  • - Paired t-test
slide13

2. Related Samples

- Paired t-test

slide14

2. Related Samples

- Paired t-test

Suppose we wanted to assess the effect of a muscle-building supplement, and randomly assign people to two groups – placebo and experimental. People differ in many characteristics (ethnicity, sex, weight, diet, etc.), and so these have been randomized across groups. Effects due to these variables are part of the “within group variance” in the denominator.

slide15

2. Related Samples

- Paired t-test

But if we give everyone the drug, and assess their performance before and after, then there is no within group variance between sample points – they are the same individual.

slide16

2. Related Samples

- Paired t-test

We can look at the distribution of differences between before and after weights, and do a “z-test” asking the question: is “0” an unusual value for this sample? Or, how likely is it that “0” (no difference) is a part of this population of differences? If “0” is unlikely, then our population is different from zero; the difference between “before” and “after” is NOT zero – there IS an effect.

slide17

2. Related Samples

- Paired t-test

Y1- 0

S/ n1

slide18

Person wt. Before wt. After Difference

1 150 156.5 1.5

2 155 156.3 1.3

3 158 159.6 1.6

4 160 161.4 1.4

5 163 164.5 1.5

6 167 166.8 -0.2

7 175 176.3 1.3

8 180 181.5 1.5

9 185 186.1 1.1

10 191 192.6 1.6

mean = 1.26

sd = 0.536

1.26 – 0

0.536 / 10

t =

= 7.46

slide19

In a two-tailed test, we are asking if a value (or sample) IS DIFFERENT FROM a sample… (it can differ because it is LARGER or SMALLER.)

slide20

In a one-tailed test, we have a preconceived hypothesis about the direction of the effect. In our experiment here, “0” should be LOWER than the mean of our differences. So, our type one error can be “pooled” into one tail of the distribution.

slide21

So, our type one error can be “pooled” into one tail of the distribution. This means we use the t value in the table A.2 corresponding to p = 0.1 to test at the p = 0.05 level.

slide23

Differences Between Means

  • A. 1 Sample
  • one sample t-test (z test)
  • B. 2 samples
  • 1. Independent Groups
  • - t-test (parametric)
  • - MWU (non-parametric)
  • 2. Related Samples
  • - Paired t-test
  • - Sign Test (non-parametric)
slide24

2. Related Samples

- Paired t-test

- Sign Test (non-parametric)

For the matched pairs, you simply record whether one partner is greater (+) or less than (-) the other:

slide25

Person wt. Before wt. After Sign

1 150 156.5 +

2 155 156.3 +

3 158 159.6 +

4 160 161.4 +

5 163 164.5 +

6 167 166.8 -

7 175 176.3 +

8 180 181.5 +

9 185 186.1 +

10 191 192.6 +

slide26

Person wt. Before wt. After Sign

1 150 156.5 +

2 155 156.3 +

3 158 159.6 +

4 160 161.4 +

5 163 164.5 +

6 167 166.8 -

7 175 176.3 +

8 180 181.5 +

9 185 186.1 +

10 191 192.6 +

Now, if we are testing the hypothesis of NO effect, then we would expect the “after” to be greater 1/2 the time (p = 0.5), and less than ½ the time (q = 0.5).

Pairs that don’t differ are dropped from the analysis, with reduction in n.

slide27

Person wt. Before wt. After Sign

1 150 156.5 +

2 155 156.3 +

3 158 159.6 +

4 160 161.4 +

5 163 164.5 +

6 167 166.8 -

7 175 176.3 +

8 180 181.5 +

9 185 186.1 +

10 191 192.6 +

Now, if we are testing the hypothesis of NO effect, then we would expect the “after” to be greater 1/2 the time (p = 0.5), and less than ½ the time (q = 0.5). So, this reduces to calculating the probability of a particular BINOMIAL OUTCOME, OR SOMETHING MORE EXTREME.

Pairs that don’t differ are dropped from the analysis, with reduction in n.

slide28

So, if p = 0.5 and q = 0.5 (no effect), what would be the probability of having at least 9/10 individuals show a weight gain just by chance?

p(9) = (p9) (q1) = 0.009760

p(10) = (p10) (q0) = 0.0009760

10!

(9!) (1!)

10!

(10!) (0!)

0.010736

This is the one-tailed probability of seeing at least 9/10 showing a weight gain (directional).

slide29

2. Related Samples

- Paired t-test

- Sign Test (non-parametric)

- Wilcoxon Signed-ranks test

slide30

2. Related Samples

- Paired t-test

- Sign Test (non-parametric)

- Wilcoxon Signed-ranks test

In the sign test, the magnitude of the difference doesn’t matter. You could have 5 big positive diffs and 4 very small diffs, and it would still be 5 + and 4 - .

slide31

2. Related Samples

- Paired t-test

- Sign Test (non-parametric)

- Wilcoxon Signed-ranks test

In the sign test, the magnitude of the difference doesn’t matter. You could have 5 big positive diffs and 4 very small diffs, and it would still be 5 + and 4 - .

The signed-ranks test takes the magnitude of the difference into account.

slide32

Example 9.5

AGGRESSION

Female w/o kittens w kittens Diff. Rank

1 3 7 4 4

2 2 8 6 6

3 5 4 -1 1.5(-)

4 6 9 3 3

5 5 10 5 5

6 1 9 8 7

7 8 9 1 1.5

slide33

Example 9.5

AGGRESSION

Female w/o kittens w kittens Diff. Rank

1 3 7 4 4

2 2 8 6 6

3 5 4 -1 1.5(-)

4 6 9 3 3

5 5 10 5 5

6 1 9 8 7

7 8 9 1 1.5

Sum ranks with positive and with negative values:

Negative = 1.5

Positive = 26.5

T = lower value = 1.5

slide34

Example 9.5

AGGRESSION

Female w/o kittens w kittens Diff. Rank

1 3 7 4 4

2 2 8 6 6

3 5 4 -1 1.5(-)

4 6 9 3 3

5 5 10 5 5

6 1 9 8 7

7 8 9 1 1.5

Sum ranks with positive and with negative values:

Negative = 1.5

Positive = 26.5

T = lower value = 1.5

slide35

N 0.05

6 0

7 2

8 4

9 6

10 8

11 11

Compare SMALLER value to critical value, at n for number of paired samples. Reject Ho if calculated value is SMALLER THAN critical value, as our is here (1.5 < 2).

slide37

Differences Between Means

  • A. 1 Sample
  • B. 2 Samples
  • C. >2 Samples
  • 1. 1 factor
  • - One way ANOVA
  • - Kruskal-Wallis
slide38

Consider 4 groups that are not normally distributed, with 10 values each.

Rank all values across categories.

Sum ranks for categories:

1 2 3 4

162.5 208.5 316.5 132.5

H = 14.273

Use Chi-square distribution,k-1 df.

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