Two ways of arriving at a conclusion 2. Inductive inference
samplepopulation samplepopulation 1. Deductive inference
Slide 3
IF YOUR DATA ARE: 1. Continuous data 2. Ratio or interval 3.
Approximately normal distribution 4. Equal variance (F-test) 5.
Conclusions about population based on sample (inductive) 6. Sample
size > 10 samplepopulation
Slide 4
Imagine the following experiment: 2 groups of crickets Group 1
fed a diet with extra supplements Group 2 fed a diet with no
supplements Weights 12.113.913.012.1 14.912.212.914.9
13.612.013.513.6 12.015.912.412.0 10.912.111.010.9 9.18.911.010.1
9.99.28.011.9 8.69.08.59.6 10.010.99.48.0 11.97.110.08.9 Mean =
12.8 Mean = 9.49
Slide 5
What youre doing here is comparing two samples that, because
youve not violated any of the assumptions we saw before, should
represent populations that look like this: 9.4912.8 Are the means
of these populations different?? Frequency Weight
Slide 6
Are the means of these populations different?? To answer this
question use a statistical test A statistical test is just a method
of determining mathematically whether you definitively say yes or
no to this question What test should I use??
Slide 7
IF YOU HAVENT VIOLATED ANY OF THE ASSUMPTIONS WE MENTIONED
BEFORE Number of groups compared 2 other than 2 T -test Direction
of difference specified? YesNo One-tailedTwo- tailed Does each data
point in one data set (population) have a corresponding one in the
other data set? YesNo Paired t-testUnpaired t-test Are the means of
two populations the same? Are the means of more than two
populations the same? Number of factors being tested 12>2 Does
each data point in one data set (population) have a corresponding
one in the other data sets? Two way ANOVA ANOVA YesNo One way ANOVA
Repeated Measures ANOVA Other tests
Slide 8
A simple t-test 1. State hypotheses H o there is no difference
between the means of the two populations of crickets (i.e. the
extra nutrients had no effect on weight) H 1 there is a difference
between the means of the two populations of crickets (i.e. the
extra nutrients had an effect on weight)
Slide 9
A simple t-test 2. Calculate a t-value (any stats program does
this for you) 3. Use a probability table for the test you used to
determine the probability that corresponds to the t- value that was
calculated. (for the truly masochistic)
Slide 10
A simple t-test 2. Calculate a t-value (any stats program does
this for you) 3. Use a probability table for the test you used to
determine the probability that corresponds to the t- value that was
calculated. DataTest statisticProbability
Slide 11
Unpaired t test Do the means of Nutrient fed and No nutrient
differ significantly? P value The two-tailed P value is <
0.0001, considered extremely significant. t = 7.941 with 38 degrees
of freedom. 95% confidence interval Mean difference = -3.307 (Mean
of No nutrient minus mean of Nutrient fed) The 95% confidence
interval of the difference: -4.150 to -2.464 Assumption test: Are
the standard deviations equal? The t test assumes that the columns
come from populations with equal SDs. The following calculations
test that assumption. F = 1.192 The P value is 0.7062. This test
suggests that the difference between the two SDs is not
significant. Assumption test: Are the data sampled from Gaussian
distributions? The t test assumes that the data are sampled from
populations that follow Gaussian distributions. This assumption is
tested using the method Kolmogorov and Smirnov: Group KS P Value
Passed normality test? =============== ====== ========
======================= Nutrient fed 0.1676 >0.10 Yes No
nutrient 0.1279 >0.10 Yes
Slide 12
Interpretation of p
X 2 = 12.52Critical value for 3 degrees of freedomat.05 level
is7.82 X 2 Table Conclusion: Probability of these data fitting the
expected distribution is p >.001
Slide 27
A little X 2 wrinkle - the Yates correction Formula is (o -e) 2
e 2 = Except of df = 1 (i.e. youre using two categories of data)
Then the formula becomes (|o -e| - 0.5) 2 e 2 =
Slide 28
A second goodness-of-fit test G-test or Log-Likelihood Ratio
Use if |o - e | < e e.g. if o is 12 and e is 7 G = 2 o ln=
4.60517 * o log 10 oeoe oeoe
Slide 29
Type of dataNumber of samples Are data related? Test to use
Nominal2YesMcNemar Nominal2NoFishers Exact Nominal>2YesCochrans
Q Summary!
Slide 30
Type of dataNumber of samplesAre data related?Test to use
Nominal2YesMcNemar Nominal2NoFishers Exact Nominal>2YesCochrans
Q Ordinal1NoKomolgorov- Smirnov Ordinal+2YesWilcoxon (paired t-test
analogue) Ordinal+2NoMann Whitney U (unpaired t-test analogue)
Ordinal+>2NoKruskal Wallis (analogue of one- way ANOVA
Ordinal>2YesFriedman two-way ANOVA All of the parametric tests
(remember the big flow chart!) have non-parametric equivalents (or
analogues)