Concepts in Inferential Statistics I

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Transcript Concepts in Inferential Statistics I

Jan 17, 2014

Hypothesis, Null hypothesis

• The probability of the alpha level.
• Research question
• Null is the hypothesis of “no
relationship”


differences between two mean
values, or a difference from a
specific level of a mean, such as
zero: hypothesis testing
• Bell curve
• Standard normal distribution

“Alpha .05”
• Expect to get results different than
those found this many times if the
experiment was repeated 100
times. Percent error we are willing
to tolerate
Student’s t-test
• Used to test for significant
Normal Distribution
has a mean of “0” and a
standard deviation of “1”
“P” Value

Confidence Intervals
• What is the expected range of
possible observations, based on
the data we have?
 Can
you sample every person in a population?
In most cases, it is impossible to survey every person in a
population to come up with a distribution of data so we never can
know for sure what a distribution may look like.
So we use a sample.
 A sample is drawn and we hope that the sample is representative of the
population. (e.g. Sheldon knocking)
 Size of sample is based in large part on the amount of variability in the
population, not the “size” of the population.
 Generally, when n greater than or equal to about 30, we assume we have a
normal distribution
 This is based in part on a concept you may recall called the “Central Limit
Theorem: the distribution of sample means is approximately normal
regardless of whether the population being sampled is normal.
 When
we talk about a “normal distribution”
you can picture the Bell Curve
 A normal distribution is defined by its mean and std. dev.
 The mean is a measure of location (where is the mean?)
 The std. dev. is a measure of spread or variation.
A
normal distribution underlying our data is a
requirement for many statistical procedures.
 Lit Eval Note: When you suspect that data may not be normally distributed,
you need to look very closely at the statistics used.

Standard Normal Curve
(Z - Distribution)
• Has a mean of “0” and a std. dev.
of “1”
• The area under the whole
distribution is 1
• The AUC between any two points
can be interpreted as the relative
frequency of the values included
between those points
• Understanding this concept is
essential to understanding p and
alpha.
Z=X–u/s

Where: X is the variable to be standardized,
u is the population mean and s is the
population standard deviation
Example: Transformation of a distribution with a mean
of 50 and a standard deviation of 5
Population Mean
μ=500
• Standard deviation
σ=100
• 68% of SAT scores
fall between 400 and
600
• Approximately 95%
of scores fall between
300 and 700
Distribution 1
Distribution 2
What happens if we do not have a normal distribution of data? Is it
possible to draw conclusions about areas under the curve, as we did
with the Z distribution, a symmetrical distribution? It depends…we often
still do, depending on how far from normal a distribution may be. Rule
of thumb, n=30.
 If we did an experiment 100 times, we would expect
to get similar results (an observed value equal to or
less than we got this time) 95 times, and we would
expect a different result only 5 times if we choose an
alpha level of .05.


The probability of concluding there is a difference
between groups when there really is no difference
between them (i.e., you found one of those times when
Sheldon knocked something other than his normal 3
times, also referred to as a type I error).
A statistical result is usually considered statistically
significant if the probability of a type I error (alpha) is
less than 5%.
 So
what happens when alpha is .0512: do
you conclude the study has obtained
non-significant results?
 What
are the implications of having
steadfast rules for statistical significance?




The level of statistical significance.
A value of p<0.05 means that the probability that the
result is due to chance is less than 1 in 20 and is the
same as alpha < 0.05.
The smaller the p-value, the greater your confidence in
the statistical result.
Alpha does not change whereas p values are
dependent on the actual value of the statistic in
question.

When we do research, we set a standard that is relatively conservative that
a researcher must meet in order to claim that s/he has made a discovery
of a some phenomenon or answered some question. The standard is the
alpha level, usually set at .05. Assuming that the null hypothesis is true,
this means we may reject the null only if the observed data are so unusual
that they would have occurred by chance at most 5 % of the time. The
smaller the alpha, the more stringent the test (the more unlikely it is to
find a statistically significant result). Once the alpha level has been set, a
statistic (like t or Z) is computed. Each statistic has an associated
probability value called a p-value, or the likelihood of an observed
statistic occurring due to chance, given the sampling distribution. Alpha
sets the standard for statistical significance, yes or no – whether or not we
can reject the null hypothesis. The p-value indicates the actual level of
how extreme the data are.
 Alpha
is the probability of a Type I error:
• The error of rejecting the null hypothesis if it is really
true – or saying something is significant when it is
not.
 We found a difference when there is not really a difference.
 With this kind of error, a drug that does not work could get to
market. Sometimes referred to as a false positive.
 Type
II error (beta):
• The probability of concluding that there is no difference
between treatment groups when there really is a
difference
• The error of accepting the null hypothesis and
concluding no difference, when it is actually false
and there is a difference.
 Failing to recognize a real difference. Sometimes referred to as
a False Negative.
 In this type error, we could keep a potentially life saving drug off
the market.
In research, we generally set, in advance of
doing research alpha (type I) levels
established at 0.05 and beta (type II) levels
at 0.20. What does this tell us?

The ability of a study to detect a significant difference
between treatment groups; the probability that a study
will have a statistically significant result (p<0.05).
Power = 1- beta (the false-negative rate). By
convention, adequate study power is usually set at 0.8
(80%). This corresponds to beta of 0.2 (a false-negative
rate of 20%). Power increases as sample size increases.
The power of a study should be stated in the methods
section of a study report.
 Statistical Power
 The probability of correctly rejecting the null hypothesis find it true when it is.
 Has become common to see power reported in clinical
studies.
 Think of it as your “confidence” in your results.
 Power is (1-Type II) error
 So, its 1 – the chance you got it wrong = the probability you got it right.
 Typically we see Alpha 0.05, Beta 0.20, Power is 80%
 While alpha = 0.05 is an absolute according to most statistical experts,
power is not.
 Power analysis is used in sample size planning and can be used for
hypothesis testing
• To calculate power all you need is:
 Desired alpha level
 An estimate of how big the effect is in the population.
 Estimate of the variability.

Suppose you calculate that you need 1800 subjects to
achieve 80% power for a hypertension study you are
conducting. After completing the study you end up
with 1788 subjects. What implications does this have
for the quality of your results? Is the study flawed
because you did not achieve sufficient power? (You
missed your goal by 12 subjects.)
• Effect size and variability of your results will dictate the power.
Depending on the size of these two variables, you may still have
reached sufficient power.
• Even if you did not reach .80 or 80%, what are the implications?
 Test
statistics take the form of:
Test Statistic = Some measure of difference
Some measure of variation
 Students
t-test is one of many test-statistics.
 The
t distribution is very similar to the Z
distribution, except that the area under the
curve is affected by the sample size.
• Z vs. t computationally:
Z = x1 – x2 / s.d.
t = x1- x2 / (s.d./sqrt n)
 One
of the most common statistical tests you
will see.
 Compares mean scores, or compares a mean score to a fixed
value:
 i.e., average change in blood pressure is more than 10 mmHg
 Average change in blood pressure between drug A and drug B is zero
 Basically, a t-test is a Z score, adjusted for sample size.
 Generally, all test statistics take on the form of “some difference
divided by some measure of variation”
 For the t-test, it is; t = mean –observation / (s / sqrt n) – however:
 The actual test statistic (the calculation) varies with the nature of your
study, depending on if you have “dependent” or “independent”
samples
 Dependent samples are also referred to as “within-subjects” “repeated
measures” or simply the same subjects are present in both groups being
compared.
 Independent samples compares two separate, unrelated groups.
 Dependence or independence of the samples affects the variability – and this is
accounted for in the calculations of the different tests (dependent and
independent t-tests)
 Look for recognition of this issue as this usually means the authors
know what they are doing!
 Assume:
• Random sample of 100 students taking statistics
exam in 2012
• Mean score=86, standard deviation=25
• From 2000 to 2012, mean score was 80 (population
mean)
 Question:
Did 2013’s students do significantly
better on the test than previous years?
 Null form: There is no difference in 2012 students test scores
than previous years.
 Assume
the variable test score has a normal
distribution
 Step 1:
 Hypotheses:
 Null Hypothesis = H0: μ equals 80
 Alternative Hypothesis = HA: μ does not equal 80 (or μ > 80?)
 In this example, should we use a 1-sided or 2-sided test?
 Step
2:
 Decide on what percent of error we are willing to tolerate (alpha)
for incorrectly rejecting our H0
 This is the Type I error - (i.e., incorrectly saying that this year’s
students did better when in fact they did not)
 Let’s say, we are willing to accept a 5% Type I error or α=0.05
 Step
3:
• Based on accepted level of α and sample size,
determine the critical value of t statistic above
which H0 will be rejected
• For α=0.05, and n=100,
• Critical value (c.v.)=1.66 (from t-distribution
table, next slide, one-sided or 1-tailed test)
 Step
4: Calculate the t statistic
t = (sample mean – population mean) / ( s / sqrt n )
t = (86 -80) / (25 / sqrt 100) = 2.4
 Step
5: Since 2.4 (t-calc) > 1.66 (t-table) the
hypothesis that the two means are equal is
rejected.
 Or, the
means are different based on this result.

Confidence interval (C.I) is an interval around our
mean that indicates the reliability of the mean

C.I. = mean ± t (s / sqrt n)

95% CI of μ = mean ± 1.96 (s / sqrt n)
 95% of all sample means fall within 1.96 standard deviations of the
population mean

99% CI of μ = x ± 2 . 58 ( s / sqrt n )
 99% of all sample means fall within 2.58 standard deviations of the
population mean1
Confidence Intervals:
 Assume:
• Sample size n=100
• Sample mean age = 54.85 years
• Sample standard deviation = 5.50
 95% CI = 54.85 ± 1.96 (5.5 / sqrt 100) = (53.78, 55.93)
• This range captures the true value of mean population age with 95%
certainty
• There is a 2.5% chance that the true mean actually lies above 55.93, or
lies below 53.78
• My question was always: “Where does the 1.96 come from?”
Short-term outcomes of an employer-sponsored diabetes
management program at an ambulatory care pharmacy clinic.
Yoder et al.
American Journal of Health-System Pharmacists