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
- Types of variables and scales
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
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)
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
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
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
Fig. 15.1
I. 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
2. Related Samples
- Paired t-test
2. Related Samples
- Paired t-test
Suppose we wanted to assess the effect of a musclebuilding 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.
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.
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.
2. Related Samples
- Paired t-test
Y1- 0
S/ n1
Person
wt. Before
wt. After
Difference
1
2
3
4
5
6
7
8
9
10
150
155
158
160
163
167
175
180
185
191
156.5
156.3
159.6
161.4
164.5
166.8
176.3
181.5
186.1
192.6
1.5
1.3
1.6
1.4
1.5
-0.2
1.3
1.5
1.1
1.6
mean = 1.26
sd =
0.536
t=
1.26 – 0
0.536 / 10
= 7.46
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.)
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.
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.
You need to look at the table,
or to enter the correct test in
SPSS
I. 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)
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:
Person
wt. Before
wt. After
Sign
1
2
3
4
5
6
7
8
9
10
150
155
158
160
163
167
175
180
185
191
156.5
156.3
159.6
161.4
164.5
166.8
176.3
181.5
186.1
192.6
+
+
+
+
+
+
+
+
+
Person
wt. Before
wt. After
Sign
1
2
3
4
5
6
7
8
9
10
150
155
158
160
163
167
175
180
185
191
156.5
156.3
159.6
161.4
164.5
166.8
176.3
181.5
186.1
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.
Person
wt. Before
wt. After
Sign
1
2
3
4
5
6
7
8
9
10
150
155
158
160
163
167
175
180
185
191
156.5
156.3
159.6
161.4
164.5
166.8
176.3
181.5
186.1
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.
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) =
10!
(p9) (q1) =
(9!) (1!)
p(10) =
10!
(10!) (0!)
0.009760
(p10) (q0) = 0.0009760
0.010736
This is the one-tailed probability of seeing at least 9/10
showing a weight gain (directional).
2. Related Samples
- Paired t-test
- Sign Test (non-parametric)
- Wilcoxon Signed-ranks test
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 - .
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.
Example 9.5
Female
AGGRESSION
w/o kittens
w kittens
Diff.
Rank
1
2
3
4
5
6
7
3
2
5
6
5
1
8
4
6
-1
3
5
8
1
4
6
1.5(-)
3
5
7
1.5
7
8
4
9
10
9
9
Example 9.5
Female
AGGRESSION
w/o kittens
w kittens
Diff.
Rank
1
2
3
4
5
6
7
3
2
5
6
5
1
8
4
6
-1
3
5
8
1
4
6
1.5(-)
3
5
7
1.5
7
8
4
9
10
9
9
Sum ranks with positive and with negative values:
Negative = 1.5
Positive = 26.5
T = lower value = 1.5
Example 9.5
Female
AGGRESSION
w/o kittens
w kittens
Diff.
Rank
1
2
3
4
5
6
7
3
2
5
6
5
1
8
4
6
-1
3
5
8
1
4
6
1.5(-)
3
5
7
1.5
7
8
4
9
10
9
9
Sum ranks with positive and with negative values:
Negative = 1.5
Positive = 26.5
T = lower value = 1.5
N
0.05
6
7
8
9
10
11
0
2
4
6
8
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).
Fig. 15.1
I. Differences Between Means
A. 1 Sample
B. 2 Samples
C. >2 Samples
1. 1 factor
- One way ANOVA
- Kruskal-Wallis
Consider 4 groups that are not normally distributed, with 10 values each.
Rank all values across categories.
Sum ranks for categories:
1
162.5
2
208.5
3
316.5
4
132.5
H = 14.273
Use Chi-square distribution,k-1 df.