ANOVA (Analysis of variance test)
Groups | Control | Group A | Group 11 | Group 111 | Group 1v | Total |
17 | 19 | 18 | 20 | 24 | ||
21 | 22 | 16 | 23 | 28 | ||
19 | 25 | 17 | 25 | 29 | ||
11 | 18 | 13 | 20 | 25 | ||
Observations | 4 | 4 | 4 | 4 | 4 | 20(n) |
Sigma X | 68 | 84 | 64 | 88 | 106 | 410 (T) |
Mean | 17 | 21 | 16 | 22 | 26.5 | |
Sigma X 2 | 1212 | 1794 | 1038 | 1954 | 2826 | 8824 |
(Signa X)2 /n | 1156 | 1764 | 1024 | 1936 | 2809 | 8689 |
Correction factor Cf = T2/N = (410) 2
------------ = 8405
20
Total sum of squares = sigma X2 - c.f = 8824 – 8405 = 419
B/N groups sum of squares = ( sigma X2)/n) - c.f = 8689 – 8405 = 284
Within groups = Total sum of square - between groups sum of squares
= 419 – 284 = 135
Degree of freedom
B/n groups = number of groups - 1 , 5-1 =4
Total degree of freedom = Total observation of all groups – 1
20 – 1 = 19
Error (df ) Total df - between groups df = 19-4 =15
Mean square(b/n groups) = B/n groups sum of squares/degree of
freedom
= 284/4 = 71
Mean square within groups = 135/15 = 9
F = Between mean squares / within groups mean square
= 71/9 = 7.89
Referring to F – ration table for (4,15) degrees of freedom we get for F = 7.89 , p greater than 0.01 , hence there is a significant difference between groups
For finding out the differences b/n the groups, error mean square Anova is made use by applying dunnets t test
T test formula
Where s2 is the error mean square obtained from Anova
Dunnets t test to determine effect of drug against control group
Statistic | A | B | C | D |
t | 1.886 | 0.472 | 2.358 | 4.481 |
D.F = 15
*P less than 0.05 , ***P less than 0.001
Therefore drug C and drug D differ significantly from control , but drug D is highly effective
Tukeys test
Steps
- Calculate an analysis of variance (e.g., One-way between-subjects ANOVA).
- Select two means and note the relevant variables (Means, Mean Square Within, and number per condition/group)
- Calculate Tukey's test for each mean comparison
- Check to see if Tukey's score is statistically significant with Tukey's probability/critical value table taking into account appropriate dfwithin and number of treatments.
Problem: Susan Sound predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides twenty-four students into three groups of eight. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with noise that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. She begins by conducting a One-way, between-subjects Analysis of Variance. She finds a significant F score. The relevant variables from her ANOVA table are:
MSwithin =4.18; M1 =6; M2 =4; M3 =3; dfwithin = 21; n = 8
EXAMPLES OF THE NULL HYPOTHESIS
A researcher may postulate a hypothesis:
H1: Tomato plants exhibit a higher rate of growth when planted in compost rather than in soil.
And a null hypothesis:
H0: Tomato plants do not exhibit a higher rate of growth when planted in compost rather than soil.
It is important to carefully select the wording of the null, and ensure that it is as specific as possible. For example, the researcher might postulate a null hypothesis:
H0: Tomato plants show no difference in growth rates when planted in compost rather than soil.
There is a major flaw with this null hypothesis. If the plants actually grow more slowly in compost than in soil, an impasse is reached. H1 is not supported, but neither is H0, because there is a difference in growth rates.
If the null is rejected, with no alternative, the experiment may be invalid. This is the reason why science uses a battery of deductive and inductive processes to ensure that there are no flaws in the hypotheses.
Many scientists neglect the null, assuming that it is merely the opposite of the alternative, but it is good practice to spend a little time creating a sound hypothesis. It is not possible to change any hypothesis retrospectively, including H0.
SIGNIFICANCE TESTS
If significance tests generate 95% or 99% likelihood that the results do not fit the null hypothesis, then it is rejected, in favor of the alternative.
Otherwise, the null is accepted. These are the only correct assumptions, and it is incorrect to reject, or accept, H1.
Accepting the null hypothesis does not mean that it is true. It is still a hypothesis, and must conform to the principle of falsifiability, in the same way that rejecting the null does not prove the alternative.
PERCEIVED PROBLEMS WITH THE NULL
The major problem with the null hypothesis is that many researchers, and reviewers, see accepting the null as a failure of the experiment. This is very poor science, as accepting or rejecting any hypothesis is a positive result.
Even if the null is not refuted, the world of science has learned something new. Strictly speaking, the term ‘failure’, should only apply to errors in the experimental design, or incorrect initial assumptions.
DEVELOPMENT OF THE NULL
The Flat Earth model was common in ancient times, such as in the civilizations of the Bronze Age or Iron Age. This may be thought of as the null hypothesis, H0, at the time.
H0: World is Flat
Many of the Ancient Greek philosophers assumed that the sun, moon and other objects in the universe circled around the Earth. Hellenistic astronomy established the spherical shape of the earth around 300 BC.
H0: The Geocentric Model: Earth is the centre of the Universe and it is Spherical
Copernicus had an alternative hypothesis, H1 that the world actually circled around the sun, thus being the center of the universe. Eventually, people got convinced and accepted it as the null, H0.
H0: The Heliocentric Model: Sun is the centre of the universe
Later someone proposed an alternative hypothesis that the sun itself also circled around the something within the galaxy, thus creating a new null hypothesis. This is how research works - the null hypothesis gets closer to the reality each time, even if it isn't correct, it is better than the last null hypothesis.
