Transcript Descriptive Statistics

Medical Statistics
S Balakrishnan
Agenda
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Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Agenda
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Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Continuous variable
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One in which research participants differ in
degree or amount.
“susceptible to infinite gradations” (p. 176, Pedhazur & Schmelkin,
1991)
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Examples: height, weight, age
Categorical variable
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Participants belong to, or are assigned to, mutual
exclusive groups
–
Nominal
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–
Used to group subjects
Numbers are arbitrary
Examples: sex, race, dead/alive, marital status
Ordinal (rank)
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Given a numerical value in accordance to their rank on the
variable
Numerical values assigned to participants tells nothing of the
distance between them
Examples: class rank, finishers in a race
Independent vs Dependent Variable
Independent
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“predictor variable”
Usually on the “x” axis
Dependent
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“outcome” variable
Usually on the “y” axis
Dependent
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Independent
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The independent variable
(a treatment) leads to the
dependent variable
(outcome)
Ultimately, we are
interested in differences
between dependent
variables
Agenda
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Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Descriptive Statistics
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These are measures or variables that summarize
a data set
2 main questions
–
–
Index of central tendency (ie. mean)
Index of dispersion (ie. std deviation)
Descriptive Statistics
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Data set for ICD
complications in 2005
14 patients
Sex: F, F, M, M, F, F, F, M,
F, M, M, F, F, F
Make: G, S, G, G, G, M,
S,S, G,G, M, S
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Central tendency is
summarized by proportion
or frequency
Sex:
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–
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Categorical data
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5/14 = .36 or 36%
9/14 = .64 or 64%
Make:
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M
F
G
S
M
6/12 = .5 or 50%
4/12 = .33 or 33%
2/12 = .17 or 17%
Dispersion not really used
in categorical data
Descriptive Statistics
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Data set SBP among a
group of CHF pts in VA
clinic
13 patients
100, 95, 98, 172, 74, 103,
97, 106, 100, 110, 118,
91, 108
Continuous variable
Central Tendency
 Mean
–
–
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Median
–
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mathematical average of all
the values
Σ (xi+xii…xn)/n
value that occupies middle
rank, when values are ordered
from least to greatest
Mode
–
Most commonly observed
value(s)
Descriptive Statistics
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Data set SBP among a
group of CHF pts in VA
clinic
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13 patients
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100, 95, 98, 172, 74, 103,
97, 106, 100, 110, 118,
91, 108
Continuous variable
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Central Tendency
 Mean
–
–
mathematical average of all
the values
Σ (xi+xii…xn)/n
= (100+95+98+172+74+103+
97+106+100+110+118+
91+108)/13 = 105.5
Descriptive Statistics
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Data set SBP among a
group of CHF pts in VA
clinic
13 patients
100, 95, 98, 172, 74, 103,
97, 106, 100, 110, 118,
91, 108
Continuous variable
Central Tendency
 Median
–
value that occupies middle
rank, when values are ordered
from least to greatest
74, 91, 95, 97, 98, 100, 100,
103, 106, 108, 110, 118,
172
 Useful if data is skewed
or there are outliers
Descriptive Statistics
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Data set SBP among a
group of CHF pts in VA
clinic
100, 95, 98, 172, 74, 103,
97, 106, 100, 110, 118,
91, 108
Continuous variable
Index of dispersion
 Standard deviation
–
–
measure of spread around
the mean
Calculated by measuring
the distance of each value
from the mean, squaring
these results (to account for
negative values), add them
up and take the sq root
Descriptive Statistics: “Normal”
Descriptive Statistics:
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Confidence Intervals
“Range of values which we can be confident
includes the true value”
Defines the “inner zone” about the central index
(mean, proportion or ration)
Describes variability in the sample from the
mean or center
Will find CI used in describing the difference
between means or proportions when doing
comparisons between groups
Altman DG. Practical Statistics for Medical Research ;1999
Descriptive Statistics:
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Confidence Intervals
For example, a “95% CI” indicates that we are
95% confident that the population mean will fall
within the range described
Can be used similar to a p-value to determine
significant differences
CI is similar to a measure of spread, like SD
As sample size increase or variability in the
measurement decrease, the CI will become more
narrow
Descriptive Statistics:
Confidence Intervals
L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5
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Prospective, randomized, multicenter trial of
different management strategies for ACS
2500 pts enrolled in Europe with 6 month followup
Primary endpoints: Composite endpoint of death
and myocardial infarction after 6 months
Descriptive Statistics:
Confidence Intervals
L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5
Descriptive Statistics:
Confidence Intervals
L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5
*Risk ratio= Riskinvasive / Risknoninvasive
When CI cross 1 or whatever designates equivalency, the p-value not be
significant.
Descriptive Statistics:
Confidence Intervals
L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5
Review
 Calculate:
–
RRR, ARR, NNT
RRR = (12.1-9.4) / 12.1 = 22%
ARR = 12.1 - 9.4 = 2.7%
NNT = 100 / ARR = 100 / 2.7 = 37
Agenda






Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Hypothesis
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
Statement about a population, where a certain
parameter takes a particular numerical value or
falls in a certain range of values.
Examples:
–
–
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A director of an HMO hypothesizes that LOS p AMI is
longer than for CHF exacerbation
An investigator states that a new therapy is 10% better
than the current therapy
Bivalirudin is not-inferior to heparin/eptifibitide for
coronary PCI
Null Hypothesis (Ho)
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“Innocent until proven guilty”
Null hypothesis (Ho) usually states that no
difference between test groups really exists
Fundamental concept in research is the concept
of either “rejecting” or “conceding” the Ho
State the Ho:
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–
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A director of an HMO hypothesizes that LOS p AMI is longer than
for CHF exacerbation
An investigator states that a new therapy is 10% better than the
current therapy
Bivalirudin is not-inferior to heparin/eptifibitide for PCI
Null Hypothesis (Ho): Courtroom Analogy
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The null hypothesis is that the defendant is
innocent.
The alternative is that the defendant is guilty.
If the jury acquits the defendant, this does not
mean that it accepts the defendant’s claim of
innocence.
It merely means that innocence is plausible
because guilt has not been established beyond a
reasonable doubt.
Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan
Agenda






Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Extrapolation of Research Findings
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
Sample population vs. the world
If your study shows that treatment A is better
than treatment B
–
–
You cannot conclude that treatment A is ALWAYS
better than treatment B
You only sampled a small portion of the entire
population, so there is always a chance that your
observation was a chance event
Extrapolation of Research Findings
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At what point are we comfortable concluding that
there is a difference between the groups in our
sample
In other words, what is the false-positive rate that
we are willing to accept
What is this called in statistical terms?
Agenda






Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Definition of p-value
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With any research study, there is a possibility
that the observed differences were a chance
event
The only way to know that a difference is really
present with certainty, the entire population
would need to be studied
The research community and statisticians had to
pick a level of uncertainty at which they could live
Definition of p-value
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This level of uncertainty is called type 1 error or a
false-positive rate
Two Types of Errors
Truth
Decision Made
Reject H0
Result
Type I Error
H0 True
Trt has no effect
Not Reject H0
Correct Decision
Reject H0
Correct Decision
Not Reject H0
Type II Error
“Power”
H1 True
Trt has an effect
Stay tuned….
Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan
Definition of p-value
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This level of uncertainty is called type 1 error or a
false-positive rate (a)
More commonly called a p-value
Statistical significance will be recognized if
p ≤ 0.05 (can be set lower if one wishes)
Trade-Off in Probability for Two Errors
There is an inverse relationship between the probabilities
of the two types of errors.
Increase probability of a type I error →
decrease in probability of a type II error
.01 .05
Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan
Definition of p-value




This level of uncertainty is called type 1 error or a
false-positive rate (a)
More commonly called a p-value
In general, p ≤ 0.05 is the agreed upon level
In other words, the probability that the difference
that we observed in our sample occurred by
chance is less than 5%
–
Therefore we can reject the Ho
Definition of p-value
Stating the Conclusions of our Results
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When the p-value is small, we reject the null
hypothesis or, equivalently, we accept the alternative
hypothesis.
–
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“Small” is defined as a p-value  a, where a = acceptable false (+) rate
(usually 0.05).
When the p-value is not small, we conclude that we
cannot reject the null hypothesis or, equivalently,
there is not enough evidence to reject the null
hypothesis.
–
“Not small” is defined as a p-value > a, where a = acceptable false (+)
rate (usually 0.05).
Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan
Agenda






Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
t-tests
Chi-square
One variable
Continuous
Categorical
Mean, SD
Frequency
One-sample t-test
Two variables
T-test
Chi-square
Three or more
variables
ANOVA
Chi-square
Two Sample Tests: Continuous Variable
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t-test
Comparing two groups, statistical significance is
determined by:
–
Magnitude of the observed difference
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–
Bigger differences are more likely to be significant
Spread, or variability, of the data

Larger spread will make the differences not significant
Two Sample Tests: Continuous Variable
Two Sample Tests: Continuous Variable
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
t-test
Comparing two groups, statistical significance is
determined by:
–
Magnitude of the observed difference

–
Spread, or variability, of the data

–
Bigger differences are more likely to be significant
Larger spread will make the differences not be significant
Key is to compare the difference between groups
with the variability within each group
Two Sample Tests: Continuous Variable
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Types t-tests
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Student t-test or two sample t-test
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Used if independent variables are unpaired
Example:
–
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A randomized trial to high dose statin versus placebo post AMI
Paired t-test
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Used if independent variables are paired
–
Each person is measured twice under different conditions
– Similar individuals are paired prior to an experiment
 Each receives a different trt, same response is measured

Example:
–
A study of ejection fraction in patients before and after Bi-V
pacing
Two Sample Tests: Continuous Variable
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t-test
Tails
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“Two-tailed”
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Most commonly used in clinical research studies
Means that the treatment group can be better or worse than
the control group
“One-tailed”
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Used only if the groups can only differ in one direction
Example: t-test
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What type of test should be
run?
How are the data related or are
they?
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Data entered into a statistical
program…
p value = 0.2329, not
significant
Agenda






Types of variables
Descriptive statistics
What is a hypothesis
Definition of a p-value
Sample vs. universe
Comparative statistics
–
–
T-tests
Chi-square
Two Sample Tests: Categorical Variables
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
Chi square (χ2) analysis
Data that is organized into frequency, generate
proportions
Based on comparing what values are expected
from the null hypothesis to what is actually
observed
Greater the difference between the observed and
expected, the more likely the result will be
significant
Chi square (χ2) analysis
Outcome
Therapy
+
-
Totals
Group A
a
b
a+b
c
d
c+d
a +c
b+d
a+b+c+d
“Control”
Group B
“Treatment”
• Null hypothesis states that outcomes of therapy A and B are equally
successful
• This is how the expected outcomes are determined
Chi square (χ2) analysis
Outcome
Therapy
Group A
+
-
Totals
a
b
a+b
c
d
c+d
a +c
b+d
a+b+c+d
“Control”
Group B
“Treatment”
• Next the actual observed values are then recorded
• With this information the χ2 value can be calculated and a p-value will
be generated
Example: χ2 analysis
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Arrange data into a 2x2 table
Treatment groups along the
vertical axis, Outcomes alone
the horizontal axis
Example: χ2 analysis

Data entered into a statistical
program
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
P-value 0.6392
Not a significant
difference
Example: Ear Infections and Xylitol
Experiment: n = 533 children randomized to 3 groups
Group 1: Placebo Gum;
Group 2: Xylitol Gum;
Group 3: Xylitol Lozenge
Response = Did child have an ear infection?
Group
Infection
1 placebo
Y
2
gum
N
3 lozenge
Y
4 placebo
N
5
gum
Y
6 lozenge
N
Count
49
150
39
129
29
137
Graduate Workshop in Statistics Session 5. Hamidieh K. 2006 Univ of Michigan
Two Sample Tests: Categorical Variables
Outcome
Therapy
+
-
Group A
a
b
c
d
“Control”
Group B
“Treatment”
(Observed  Expected)
 = 
Expected
allCells
2
2
Example: Ear Infections and Xylitol
Infection
Yes
Group
Placebo Gum
Xylitol Gum
Xylitol Lozenge
Total
Count
Expected Count
Count
Expected Count
Count
Expected Count
Count
Expected Count
49
39.1
29
39.3
39
38.6
117
117.0
No
129
138.9
150
139.7
137
137.4
416
416.0
Total
178
178.0
179
179.0
176
176.0
533
533.0
Compute expected count for each cell:
Expected count = (Row total)  (Column total) / Total n
Example: 39.1 = (178 × 117) / 533
Or intuitively, calculate overall infection rate
= total number infected / total number = 117/533 = .2195
Now, assuming no difference between treatments, the infection rate will be
the same in each group
= .2195 x total for each group = .2195 x 178 = 39.1
Graduate Workshop in Statistics Session 5. Hamidieh K. 2006 Univ of Michigan
Example: Ear Infections and Xylitol
Infection
Yes
Group
Placebo Gum
Xylitol Gum
Xylitol Lozenge
Total
Count
Expected Count
Count
Expected Count
Count
Expected Count
Count
Expected Count
49
39.1
29
39.3
39
38.6
117
117.0
No
129
138.9
150
139.7
137
137.4
416
416.0
Total
178
178.0
179
179.0
176
176.0
533
533.0
→ From a table, p = 0.035
Graduate Workshop in Statistics Session 5. Hamidieh K. 2006 Univ of Michigan
Conclusion
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There are many ways to describe one’s data
P-values are the maximum acceptable false
positive rate
Remember the Courtroom Analogy when it
comes to the Null hypothesis
Choice of statistical test depends on type of
variable and number of comparison groups
References
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Neely JG, et al.
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–
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Laryngoscope, 112:1249–1255, 2002
Laryngoscope, 113:1534–1540, 2003
Laryngoscope, 113:1719 –1724, 2003
Guyatt G, et al. Basic Statistics for Clinicians.
CMAJ. 1/1/95
http://www-personal.umich.edu/~khamidie/?M=A
Altman, DG. Practical Statistics for Medical
Research. 1999.
Thank you