HTHSCI 2S03 Lecture Notes - Lecture 6: Null Hypothesis, Cadency, Random Variable
26 May 2018
School
Department
Course
Professor
2S03: Sessions 6 - 9
One-Way ANOVA, Multiple Comparisons Procedures
One-Way ANOVA
• Looks at one- factor
• Developed by Ronald Fisher (British statistician)
• used to compare means of three or more groups of continuous numbers
• extension of independent t test (for two groups)
• doest look at diffeees i pais of goup eas as i a t-test) due to high error
• The means of 3 or more groups
• Eo ieases eah t test hih is h e dot do ultiple tests fo the
• instead, it compares the spread (variance) in group means to the spread (variance) you’d
expect in those means if all the groups were sampled from a single population (i.e., no true
differences between the groups)
• the comparison is expressed in a test statistic called the F statistic, the ratio of the variability
between groups to the variability within groups
o is there true differenced between the means? Or are they probably taken from the
same population?
Analysis of Variance Background: Why not use t-test ?
• We already know how to test the null hypothesis of no difference between two population
means
• Question: How do we test the null hypothesis of no difference among several population
means?
• Why not use a t-test for each pair of means?
• Suppose there are 5 populations involved. Then the # of possible pairs of sample means is 10.
So, why not do 10 separate t-tests?
• 10 separate T tests back to back with an error of 0.05 will give an alpha of 40%, instead, we do
one test to see if there is a difference between the mean differences
Analysis of Variance -Issue of type I error
• In this example, because H0 is correct, rejecting H0 means committing a type I error of =0.40!
• Thus, an important consequence of performing all possible t-tests is that it is highly likely to lead
to a false conclusion due to a high type I error.
• Question: Is there a more efficient & less error prone technique for testing the differences
among several means?
o F test: T test ^2
One Way ANOVA
• there is one source of variation, or factor (with several levels)
• F test used in one-way ANOVA is same as independent t test if there are only two groups
find more resources at oneclass.com
find more resources at oneclass.com
• F statistic is square of t-statistic (F & t distributions related)
• major advantage of F test: applies to more than two groups
One Way ANOVA Assumptions
One-way ANOVA for 3+ independent groups has similar assumptions to those applying to independent
t-tests (for 2 groups):
1. Samples for all groups are randomly selected from the population
2. Populations from which samples are selected are normally distributed
3. Population standard deviations are the same for all groups
Lecture Question 2 (Session 6)
The ANOVA procedure is a statistical approach for determining whether or not:
1. The means of two samples are equal
2. The means of two or more samples are equal
3. *The means of more than two samples are equal
4. The means of two or more populations are equal
3+ samples, do the samples come from the same population?
One Way ANOVA
two methods used to calculate the F statistic:
• Fishes ethod oe ople alulatios, logi of test is lea
• Mean Sum of Squares (simple calculations, logic of test less clear)
• We show both methods, to ensure an understanding of ANOVA and the role of the F statistic
Method 1: Roald Fisher’s Method
• How can we choose between these competing explanations?
• Compare the variance of sample means to the variance of sample means drawn from a single
distribution:
F = (variance of sample means)
(mean of sample variances)/n
Variance of sample: how spread our or different are the means of the 3 groups compared to each other,
spread across the 3 group means
find more resources at oneclass.com
find more resources at oneclass.com
Group 1 average – aeage of all goups eas +…
Mean of sample variances: average of the goups variances
a calculated F value near 1.0 suggest the data came from a single population with a relatively large
variance
Average of 3 variances
N = total number of groups
One-Way ANOVA
• Since F is > 1.0, the data suggest that the difference in group means may be due to the 3
samples coming from different populations rather than from a single population
• the differences in group means may reflect a treatment effect, not random variation within a
single population
• Fisher created the F distribution in order to have a more rigorous statistical test for determining
when the F statistic is significant (suggesting group differences are coming from different
populations)
One-Way ANOVA
• there is a family of F distributions
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Then the # of possible pairs of sample means is 10. Analysis of variance -issue of type i error. Method 1: ro(cid:374)ald fisher"s method: how can we choose between these competing explanations, compare the variance of sample means to the variance of sample means drawn from a single distribution: F = (variance of sample means) (mean of sample variances)/n. Variance of sample: how spread our or different are the means of the 3 groups compared to each other, spread across the 3 group means. Group 1 average a(cid:448)e(cid:396)age of all (cid:1007) g(cid:396)oup(cid:859)s (cid:373)ea(cid:374)s + . Mean of sample variances: average of the (cid:1007) g(cid:396)oup(cid:859)s variances a calculated f value near 1. 0 suggest the data came from a single population with a relatively large variance. Degrees of freedom: k 1/n- k: k-total number of groups, n- total number of observations over the 3 groups.
