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Making Inferences, AKA
Hypothesis Testing
Assignment 2 and 3
• You should have received feedback on
assignment 2.
– Great job everyone.
• Please send everything to both me and
Lamiya.
• Mofiz Haque please stop by I have a
question about your email address.
• Assignment 3 is assigned today.
So Far
• You know how to describe variables:
– Conceptually with a taxonomy
– Graphically
– Numerically
• You know how to describe some distributions:
– Empirically
– Theoretically
• You have been exposed to two statistical packages
to help you do these tasks:
– R with Rcmdr
– SAS Enterprise Guide
So Far
• Probability is scored between 0 and 1.
0
0.5
1
Impossible
As likely as not
Certain
Unlikely to occur
Likely to occur
• Area under a curve or heights of bars
represent probability.
From Last Time
• I talked about when a variable (really its
distribution) is (theoretically) normally
distributed, it is described by only two
parameters (the first two moments of the mean),
the mean and standard deviation.
• When you are taking sample means (with more
than one observation in the mean) and you plot
the means, the density looks normally
distributed. This fact that the sampling
distribution of means looks normal (irrespective
of the original distribution) is called the Central
Limit Theorem.
Moving On
• The next steps are to describe other types
of distributions and figure out how to
quantify just how unusual a weird statistic
from your sample actually actually is.
• You are not always going to be making
generalizations about comparing means.
– Comparing variability (variance) is hugely
important.
Variability of Sample Means
• Recall that the number of people
(observations) in each sample mattered a
lot in determining whether the sampling
distribution looked normal.
– If you have a decent size sample (the number
of people in each sample), it is hard to get
very extreme values out of a normal sampling
distribution because the extremely big values
tend to cancel out the extremely small values.
1500
0 500
Frequency
Actual Scores
300
400
500
600
700
600
700
600
700
scores
600
200
0
400
500
bunchOfMeans
200
600
Bunch of Means sample N = 20
0
The distribution of
the means from
sample size of 20 is
narrower still (and
bell-shaped).
300
Frequency
The distribution of
the means from
sample size of 5 is
narrower than the
original values (and
bell shaped).
Frequency
Bunch of Means sample N = 5
300
400
500
bunchOfMeans20
Variability Between Samples
• The width of the sampling distribution of
the means got narrower and narrower as
the size of each of the samples increased.
• The variability within a sample (of size 1)
is called the standard deviation.
• The variability across the means when you
have samples bigger than size 1 is called
the standard error.
Standard Error
• The formula for the
standard error of the
means is just the sample
standard deviation
formula with a tweak to
indicate the impact of the
sample size.
• The SE plays a huge role
in all inferences. You
need it to determine what
is an odd sample.
SD
SEMean 
Sample Size
Standard Error Formula
• As you move through the year, you will meet
many formulas for standard errors.
– If you are testing to see if there is a difference
between two groups, you use a slightly different
formula.
– If you are working with the distribution of counts of
events happening or not happening in many trials
(yes/no getting pregnant on many attempts), the SE
formula is different but it plays the same role. It helps
you determine what is an unusual value.
Probability Functions
• Some people are entertained while others
are horrified at the prospect of having to
do calculus to figure out the area under
the curve corresponding to what made an
unusual sample. Happily, you don’t have
to. You can use the probability functions in
a language like SAS or R.
Quantiles
• Say you want to know what quantile
corresponds to a standard normal value.
• The standard normal is where you have
rescaled your values so they are measured with
a mean of 0 and standard deviation of 1.
thingy ~ N (0,1)
• For example, you may want to know what value
cuts off the most extremely large 5% of a
standard normal curve.
Z-scores for Percentages
Z-scores for Percentages
What percentiles?
• You are far more likely to want to know
what percentile your actual scores
correspond to. To get those values, you
will use the CDF function (Cumulative
Density Function).
-2
0.0
0.2
-1
0.6
-3
1
0.8
2
1.0
-3
-2
-2
-1
-1
0
0
z
1
1
2
2
3
0
Quantile (Z)
0.4
Probability
-4
z
3
-3
0.0
-2
0.2
-1
0
z
0.4
p
1
0.6
2
0.8
3
1.0
0.0
0
100
150
0.1
0.2
0.3
Probability density
50
frequency
0.4
Null Hypothesis
• When you design an experiment, you typically
propose a hypothesis indicating that nothing
interesting is going on.
– For example, if you expect a drug and a placebo to
act the same way, your null hypothesis is that the
average difference is 0.
– You reject the null hypothesis if your sample is too far
out in the tails of the null distribution.
– You typically set up this target (dummy hypothesis)
and hope your data does not look like this.
Hoe Hoe Hoe
• The null hypothesis is typically written H0.
That is pronounced H-zero or H-not. Don’t
call it “hoe”.
• The alternative hypothesis is typically
written H1 or HA.
What could possibly go wrong?
• When you do an experiment you come up with a
hypothetical population mean and SD and have
a computer calculate sampling distribution of the
means (for your sample size). You can then test
to see if your data is compatible or weird giving
the population mean and standard error.
• Call this distribution “the null distribution”
because it is what you expect and nothing
interesting is going on if you find it is true.
• What could possibility go wrong?
What could possibly go wrong?
• Your guess at the population mean was right but you
could get a sample by chance (poor luck) that was from
way out in the tails of the distribution.
– The first thing that could go wrong is called the Type I (one)
Error.
• Things could be really bad and your guess about the
population mean was wrong but you get a sample that is
compatible with your original hypothesis that is not in
agreement with reality (this 2nd thing that could go wrong
is called the Type II (two) Error.
– You won’t notice that the distribution is actually centered around
an alternative mean and has an alternative distribution.
Think of…
Pascal’s Wager
The TRUTH
Your Decision
God Exists
God Doesn’t Exist
BIG MISTAKE
Correct
Correct—
Big Pay Off
MINOR MISTAKE
Reject God
Accept God
Type I and Type II Error in a Box
Your Statistical
Decision
Reject H0
True State of Null Hypothesis
H0 True
H0 False
Type I error (α)
Correct
Correct
Type II Error (β)
Do not reject H0
Analogy to Quality Control
• In my humble opinion, people typically
worry too much about the Type I error.
The probability that this error happens is
called the p-value and this is called the α
(alpha) level.
• Failing to realize that the data should be
described by an alternative distribution is
called the β (beta) error.
Hypothesis Testing Analogies
Power 1- b
Reject Null
Is a real difference
Is no real difference
No Error (true positive)
Type 1 error
Type 2 error
No Error (true negative)
Fail to reject
b
Low metastasis potential
Sensitivity
High PSA
Normal PSA
1- a
Is a really caner
No cancer
No error (true positive)
False positive
False negative
No error (true negative)
Specificity
Highly aggressive breast cancer
Positive image
Negative
a
Is a really caner
No cancer
No error (true positive)
False positive
False negative
No error (true negative)
A Tale About Two Tails
• If you want to test to see if your data is
incompatible with a null hypotheses, you specify
just how weird it needs to be to be called weird.
That is, you specify the alpha level. Typically
you say a sample statistic that could happen 1 in
20 times is too uncommon to say it happened by
chance alone.
• For example, you have a hypothetical mean and
if your sample mean is very high or very low
relative to it, you say it is too odd and you reject
the null hypothesis.
• Using the code from earlier in the lecture, you
could figure out the probability of a value.
One-Tailed
• Typically you want to know if your value differs
from the population value. In other cases (very
rarely), you may be interested if and only if the
value is greater than the population value. In yet
other cases (very rarely), you may be interested
if and only if the value is less than the population
value.
• The test of a difference is a two-tailed test
because the value could be unusually high or
low. The test of “more than” (as opposed to
“different”) is a one-tailed test. The test of “less
than” is also a one tailed test.
Splitting Tails
• If you do a two-sided test and you say a sample
is odd if it occurs only 1/20 times, you need to
split that .05 percent of the weirdness into both
tails. So you cut the distribution such that a
sample which is in the upper .025 or lower .025
of the distribution is grounds for rejecting the null
hypothesis. But if you say that you are only
interested in whether this sample is greater than
the hypothetical mean, you can shove all .05 into
one tail and it is relatively easy to find a weird
sample.
Some Moron Tails…
• The inexact use of Fisher's Exact Test in six
major medical journals by McKinney et al., JAMA
Vol. 261 No. 23, June 16, 1989
– We reviewed the use of Fisher's Exact Test in 71
articles published between 1983 and 1987 in six
medical journals. Thirty-three of 56 selected articles
did not specify use of a one- or two-tailed test, and 12
(36%) of these actually used the one-tailed test. Five
(42%) of these 12 articles contained at least one table
in which the standard significance level of P less than
.05 was no longer met when a two-tailed analysis was
run instead.
Extreme Caution
• If an outcome could biologically be either above
or below a population mean, do the two sided
tests. There are terrifying scenarios that begin
with a standard of care that is so thought to be
so good that a new (less invasive) treatment
could only be worse. So a researcher does a
one-sided test to see if the new treatment is
worse. In reality, the gold standard is harmful
(pure oxygen to neonates). Therefore, you do
not see a statistically significant difference. In
other words, they would fail to see the harmful
effect of the treatment as statistically significant.
What could possibly go wrong?
• Recall that in addition to the Type I error
caused by having an unusual sample that
really came from the null distribution, you
could get a value from the alternate
distribution that was compatible with the
null hypothesis.
Alpha and Beta
• Alpha and Beta errors are intimately
connected in testing hypotheses and van
Belle does not make this clear enough. An
alternate presentation can be found in
Normal and Streiner’s Biostatistics: The
Bare Essentials. If you are math phobic, I
highly recommend the book.
Blood Sodium Example
• The story begins with a measure of blood
sodium with a known population mean of 140
mmol/L and a standard deviation of 2. In the
study, blood measures are taken on 25 people
and the mean is 137.5. The question is “does it
look unlikely that the sample mean came from a
population with a mean of 140 or do you want to
conclude that the true population mean is
different?”
• What do you do?
Steps to the Comparison
• What is the standard error?
• How many standard errors away from the mean
is this sample?
• If testing for a difference between the groups at
the alpha .05 level, what is the cut point in zunits?
• What is the cut point in the original units?
• What is the power?
• What is the beta error?
• What happens if you use a smaller sample?
The SE
• The Standard Error of the mean:
• The Z score:
( x  )
z  /
n
SEMean 
SD
Sample Size
Calculating a Z Score
It is a darn unusual sample if the population mean is 140.
The Actual Cut Point
136
138
140
142
0.0
0.2
0.4
Density
0.6
0.8
Sample size = 25
• What happens when your sample size was
smaller?
Sample size = 4
Running the Analyst
Pick Your Study Design
Fill in the Blanks
Get Results as a Table
…or as a picture
Other Software Packages
• S-Plus can easily
produce information
on power:
Best Guesses
• So far I talked about making judgments
regarding when a sample is compatible
with a distribution. Another very important
task is making a guess about a population
value and specifying the precision of your
guess.
• This is the process of building confidence
intervals.
100% Confidence Intervals
• Say I do a sample of ages of Stanford
undergraduates. My mean from the sample is
20 years old. That is my point estimate of the
population mean. I know that the true mean is
not exactly 20. So I give myself some wiggle
room by saying 20 plus or minus something.
That range is called the confidence interval.
• I want to be 100% certain that my guestimated
range includes the true population mean, so I
say age 20 +/- 90 years.
Can I do better than that?
• The true population mean is going to be within the range
of 0 and 110 years old. So, I have built a 100%
confidence interval.
• Say I get a sample of 25 undergrads, calculate their
mean age, add +/- 10 years and call that the confidence
interval. The population mean will or will not be inside of
the range. So reality is either yes or no. How do I
specify a probability here?
• You want specify a range that when you do the sampling
experiment many times, you will usually capture the true
value within the guestimated range. That is the typical
definition of a confidence interval.
• You use the sampling distribution we have been talking
about to calculate those values.