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Universita’ degli Studi di Milano
Corso di Laurea Magistrale in Farmacia
Aspetti di economia e marketing dei medicinali e medicinali generici
- Modulo: Medicinali generici
Prof. Andrea Gazzaniga
Richiami di Statistica
Dott. Matteo Cerea
Basic statistics
Dr. Matteo Cerea, PhD
Why Statistics?
Two Purposes
1. Descriptive
Finding ways to summarize the
important characteristics of a
dataset
2. Inferential
How (and when) to generalize from a
sample dataset to the larger
population
Descriptive
Statistics
Provides graphical and numerical ways to
organize, summarize, and characterize a
dataset.
VARIABLE
A characteristic or a property that can vary
in value among subjects in a sample or a
population.
Ex:
• The weight of tablets in a batch
• The concentration of drug in plasma in
patients after the administration of a
fixed dose
Types of Variables
Predictor variable:
The antecedent conditions that are going to be used to
predict the outcome of interest.
If an experimental study, then called an “independent
variable”.
x
Outcome variable:
The variable you want to be able to predict.
If an experimental study, then called a “dependent variable”.
y=f(x)
Continuous variable:
Can assume an infinite number of possible values that fall
between any two observed values: the lowest and the highest.
Ex: the drug content of tablets in a batch expressed as microgr
Ranked variable:
Are continuous variables although they do not represent physical
measurement, such scale represent numerically ordered system.
Ex:
0
1
2
3
4
5
no encrustation
microscopic deposits on <50% of the stent
microscopic deposits on >50% of the stent
small macroscopic deposits on <50% of the stent
small macroscopic deposits on >50% of the stent
heavy macroscopic deposits
Discrete (discontinous, meristic) variable:
Consists of separate, indivisible categories. Discrete variables
have integer numbers
Ex:
# of asthma attacks,
# of fatalities,
# of colonies of microrganisms
Nominal variable (categorical):
Cannot be measured because of their qualitative nature
Ex: sex, gender, side effects associated with the treatment
Nominal ranked (ordinal) variables
Ex: side effects associated with the treatment, if ordered
How to present data?
1.Describing data with tables and graphs
(quantitative or categorical variables)
2.Numerical descriptions of center, variability, position
(quantitative variables)
3.Bivariate descriptions
(in practice, most studies have several variables)
1. Tables and Graphs
There are several types of graphs or plots employed to
display scientific data:
• Graphs or plots that are employed to describe
relationships between a fixed (independent) variable
and a dependent variable
• Graphs that are employed to pictorially describe
distributions of data
Frequency distribution lists possible values of variable
(or intervals) and number of times each occurs
Example
Pharmaceutical Statistics, David Jones, Pharmaceutical Press
http://books.google.co.ve/books?id=oXZD1GPOJIcC&printsec=frontcover&hl=it&sou
rce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
Frequency Tables
Frequency Distribution
Histogram: Bar graph of frequencies
or percentages or relative
frequency
Frequency Distribution
Frequency Distribution
309,1-310,0
308,1-309,0
307,1-308,0
306,1-307,0
305,1-306,0
304,1-305,0
303,1-304,0
302,1-303,0
301,1-302,0
300,1-301,0
299,1-300,0
298,1-299,0
297,1-298,0
296,1-297,0
295,1-296,0
294,1-295,0
293,1-294,0
292,1-293,0
291,1-292,0
290,1-291,0
Relative Frequency Distribution
Proportion of tablets in Interval
0,180
0,160
0,140
0,120
0,100
0,080
0,060
0,040
0,020
0,000
Cumulative frequency distribution data
Less than
More than
Cumulative frequency distribution graph
Less than
More than
Qualitative or nominal
Civil status X : disconnected qualitative o determined on nominal scale
Ex. 4 different possibilities
x1 = N x2 = C x3 = V x4 = S
xi : k values of X
ni : frequency of xi
n: total number of observations
fi = ni /n relative frequency; pi percent frequency
Distribution of frequency of X is
xi
ni
N
C
V
S
6
7
4
3
n=20
fi = ni /n pi = fi · 100%
0.30
0.35
0.20
0.15
30
35
20
15
1.00
100
Pie chart
Ex. Annual income in thousand Euro W : continuous quantitative
Data (k = 20) are devided in classes (4)
ai : wideness (amplitude) of each class
li = ni /ai : frequency density
Frequency table:
xi
ni
fi
Ni
ai
li
40 ⊣ 50
50 ⊣ 58
58 ⊣ 70
70 ⊣ 95
3
6
4
7
20
0.15
0.30
0.20
0.35
1.00
3
9
13
20
10
8
12
25
0.30
0.75
0.33
0.28
2. Descriptive Measures
Numerical descriptions
Let X denote a quantitative variable, with
observations X1 , X2 , X3 , … , Xn
• a-Central Tendency measures.
They are computed to give a “center” around
which the measurements in the data are distributed.
• b-Variation or Variability measures.
They describe “data spread” or how far away the
measurements are from the center.
• c-Relative Standing measures.
They describe the relative position of specific
measurements in the data.
a. Measures of Central Tendency
• Mean (average):
Sum of all measurements divided by the number
of measurements.
N is the number of
observations
• Weighted mean:
Each datum point does not contribute proportionally
w is the frequency
a. Measures of Central Tendency
• Median:
The central number of a set of data arranged in
order of magnitude.
• Mode:
The most frequent measurement in the data.
Calculation of the mean
= (2+0+0+6+4+24+9+6+1+0)
10
Calculation of the median
0, 0, 0, 1, 2, 4, 6, 6, 9, 24
3
= 5.2
Properties of mean and median
• For symmetric distributions, mean = median
• For skewed distributions, mean is drawn in direction
of longer tail, relative to median
• Mean valid for continuous scales, median for
continuous or ordinal scales
• Mean sensitive to “outliers” (median often preferred
for highly skewed distributions)
• When distribution symmetric or mildly skewed or
discrete with few values, mean preferred because
uses numerical values of observations
Ex. Tmax, median (range); AUC, mean (std.dev)
In other words…
• When the Mean is greater than the Median the
data distribution is skewed to the Right.
• When the Median is greater than the Mean the
data distribution is skewed to the Left.
• When Mean and Median are very close to each
other the data distribution is approximately
symmetric.
b. Describing variability
Range: Difference between largest and smallest
observations
(but highly sensitive to outliers, insensitive to shape). Used
for non-normally distributed data (Ex: tmax)
Mean deviation: The average distance from the mean
The deviation of observation j from the mean is
yj-y
Mean deviation: The average distance from the
mean
MD:
(𝑋𝑗−𝑋𝑚)
𝑁
# drug content
Absolute values of errors
1
100,6
0,1
2
98,3
2,2
3
98,9
1,6
4
95,1
5,4
5
104,5
4,0
6
105,5
5,0
mean
100,5
sum
18,3
MD
3,1
Variance: is the “sums of squares” (SS) of the sample
( yi  y ) ( y1  y )  ...  ( yn  y )
s 

n 1
n 1
2
2
2
2
• It is a measure of “spread”: the larger the deviations
(positive or negative) the larger the variance
The variance of a sample is the “sums of squares” (SS)
𝑆𝑆 =
𝑌𝑗 − 𝑌
2
The mean sums of squares
σ 2=
𝑌𝑗−µ 2
𝑁
The variance of a sample of a population
s 2=
𝑌𝑗 −𝑌 2
𝑁−1
The variance of a sample is the “sums of squares” (SS)
𝑆𝑆 =
𝑌𝑗 − 𝑌
2
The mean sums of squares
σ 2=
𝑌𝑗−µ 2
𝑁
Population
The variance of a sample of a population
s 2=
𝑌𝑗 −𝑌 2
𝑁−1
Sample
The variance of a sample is the “sums of squares” (SS)
( yi  y ) ( y1  y )  ...  ( yn  y )
s 

n 1
n 1
2
2
2
2
Standard deviation: s is the square root of the variance,
s 
s
2
• It is a measure of “spread”: the larger the deviations
(positive or negative) the larger the variance
The variance of a population is
( yi  my ) ( y1  y )  ...  ( yn  y )
s 

n n 1
n 1
2
2
2
2
2
s
The standard deviation s is the square root of the variance,
ss 
2
2
ss
Ex: sample of a population: 100 tablets (sample) are removed from a batch of
1.000.000 tablets (population) and tested.
The variance of a single random sample of measurements does not provide a
good estimation of variance of the population from which the sample was derived
A good estimation of population variance can be achieved from sample data if an
average of several sample variance is calculated
Concentration of a penicillin antibiotic in 5 bottles
Bottle #
Concentration of penicillin (mg/5mL)
Contribute
to the
variance
1
125
134,56
2
124
123,21
3
121
92,16
4
123
112,36
5
16
1840,41
mean (mg/ 5mL)
Total variance (sample)
101,8
2302,7
standard deviation
48,0
median
123
Standard deviation (error) of the mean: SEM
Concentration of amoxicillin in 5 aliquots tested 5 times
N = observation in sample
Aliquot 1
mean
s
Aliquot 2
Aliquot 3
Aliquot 4
Aliquot 5
25,1
27,6
24,3
23,9
25,7
25,4
25,5
26,4
24,9
23,5
21,9
25,6
25,1
26,1
24,2
24,5
25
27,1
27
25,7
23,1
24,2
25
25,2
24,3
24,0
25,6
25,6
25,4
24,7
1,5
1,3
1,1
1,2
1,0
Mean total
25,1
SEM
0,70
Standard deviation (error) of the mean: SEM
SEM = s/ 𝑁
s = standard deviation of the sample
N = # of observations of the sample
Standard deviation (error) of the mean: SEM
Concentration of amoxicillin in 5 aliquots
Aliquot 1
mean
s
Aliquot 2
Aliquot 3
Aliquot 4
Aliquot 5
25,1
27,6
24,3
23,9
25,7
25,4
25,5
26,4
24,9
23,5
21,9
25,6
25,1
26,1
24,2
24,5
25
27,1
27
25,7
23,1
24,2
25
25,2
24,3
24,0
25,6
25,6
25,4
24,7
1,5
1,3
1,1
1,2
1,0
Mean total
25,1
SEM
0,70
SEM estimed
0,66
Standard deviation of a sample
is an estimation of the variability of a population
the value does not reduce if the # of observations
increases
Standard deviation of the mean
is an measure of the variability (precision) of the estimation
of a defined population parameter (i.e. the mean)
As the size of the sample increases, the magnitude of the
standard error decreases
Coefficient of variation (CV)
𝑆
CV (%) = x 100
𝑋
Accuracy: the closenes of a measured
value to the true value
(the value in absence of error)
Absolute error: error abs = O - E
E (true value, exact)
O (observed value, or mean)
Relative error:
error rel = error abs = O – E
E
E
Precision: describes the dispersion
(variability) of a set of measurements
Typically precision is associated with low
dispersion of the value around a central
value (low standard deviations)
Accuracy: Accuracy is how close a measurement is to the "true"
value. In a laboratory setting this is often how far a measured value
is from a standard with a known value that was measured by
different technology or on a different instrument.
Precision: Precision is how close repeated measurements are to
each other. Precision has no bearing on a target value, it is simply
how close multiple measurements are together. Reproducibility is
key to scientific research and precision is important in this aspect.
c. Measures of position
pth percentile: p percent of observations below it,
(100 - p)% above it.
• Example, if in a certain data the 85th percentile is 340 means that
15% of the measurements in the data are above 340. It also means
that 85% of the measurements are below 340
• Notice that the median is the 50th percentile
 p = 50: median
 p = 25: lower quartile (LQ)
 p = 75: upper quartile (UQ)
 Interquartile range IQR = UQ - LQ
Quartiles portrayed graphically by box plots
(John Tukey)
Example: weekly TV watching for n=60 from student survey data file, 3 outliers
Box plots have box from LQ to UQ, with median marked.
They portray a five-number summary of the data:
Minimum, LQ, Median, UQ, Maximum
except for outliers identified separately
Outlier = observation falling
below LQ – 1.5(IQR)
or
above UQ + 1.5(IQR)
Ex. If LQ = 2, UQ = 10, then IQR = 8 and outliers above 10 +
1.5(8) = 22
Normal distribution curve
Normal distribution curve
In a normal distribution of data, also known as a bell
curve,
- the majority of the data in the distribution
approximately 68% will fall within plus or minus one
standard deviation of the statistical average.
This means that if the standard deviation of a data set is 2, for
example, the majority of data in the set will fall within +2 and -2 the
average.
-95.5% of normally distributed data is within two standard
deviations of the mean,
-over 99% are within three
For any data
• At least 75% of the measurements differ from the mean
less than twice the standard deviation.
• At least 89% of the measurements differ from the mean
less than three times the standard deviation.
Inferential
Statistics
…mathematical tools that permit the
researcher to generalize to a population
of individuals based upon information
obtained from a limited number of
research participants (observation)
Sample statistics /
Population parameters
• We distinguish between summaries of samples
(statistics) and summaries of populations (parameters).
• Common to denote statistics by Roman letters,
parameters by Greek letters:
Population
mean =m,
standard deviation = s
proportion  p
In practice, parameter values unknown, we make
inferences about their values using sample statistics.
Sample proportion
Definition. The statistic that estimates the
parameter 𝜋, a proportion of a population
that has some property, is the sample
proportion p
𝑝=
𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒
The sample mean X estimates the population mean m
(quantitative variable)
The sample standard deviation s estimates the
population standard deviation s (quantitative
variable)
A sample proportion p estimates a population
proportion π (categorical variable)
Ex.
From a population of n individuals 2 samples of 100 individuals are
extracted.
mean height of the first sample X1 = 168 cm
mean height of the first sample X2 = 162 cm
…
until all the samples are estracted.
The sample proportion estimates the mean of
population but with uncertainty.
The uncertainty dependes upon:
1 - sample dimension
2 - variability of the population
the means are different, but how are they
distributed?
population
sample n1
sample n2>n1
Standard deviation of sampling distribution
Standard Error
SE = s/ 𝑁
SEM = s/ 𝑁
𝑝(1 − 𝑝)
𝑛
4. Probability Distributions
Probability: With random sampling or a randomized
experiment, the probability an observation takes a
particular value is the proportion of times that
outcome would occur in a long sequence of
observations.
Usually corresponds to a population proportion (and
thus falls between 0 and 1) for some real or
conceptual population.
A probability distribution lists all the possible values and their
probabilities (which add to 1.0)
Basic probability rules
Let A, B denotes possible outcomes
•
•
•
•
P(not A) = 1 – P(A)
For distinct (separate) possible outcomes A and B,
P(A or B) = P(A) + P(B)
If A and B not distinct,
P(A and B) = P(A) x P(B given A)
For “independent” outcomes, P(B given A) = P(B),
so P(A and B) = P(A) x P(B).
Probability distribution
of a variable
Lists the possible outcomes for the “random variable” and
their probabilities
Discrete variable:
Assign probabilities P(y) to individual values y, with
0  P( y)  1, P( y)  1
In practice, probability distributions are often estimated from
sample data, and then have the form of frequency distributions
Like frequency distributions, probability distributions have
descriptive measure like mean and standard deviation
m  E (Y )   yP( y)
Expected
value
Standard Deviation - Measure of the “typical” distance of an
outcome from the mean, denoted by σ
If a distribution is approximately bell-shaped, then:
• all or nearly all the distribution falls between
µ - 3σ and µ + 3σ
• Probability about 0.68 falls between
µ - σ and µ + σ
•
•
•
•
Continuous variables:
Probabilities assigned to intervals of numbers
Most important probability distribution for continuous
variables is the normal distribution
Symmetric, bell-shaped
Characterized by mean (m) and standard deviation (s),
representing center and spread
Probability within any particular number of standard
deviations of m is same for all normal distributions
An individual observation from an approximately normal
distribution has probability
 0.68 of falling within 1 standard deviation of mean
 0.95 of falling within 2 standard deviations
 0.997 of falling within 3 standard deviations
The normal curve is often called the
Gaussian distribution, after Carl
Friedrich Gauss, who discovered
many of its properties. Gauss,
commonly viewed as one of the
greatest mathematicians of all time,
was honoured by Germany on their
10 Deutschmark bill.
Properties (cont.)
• Has a mean = 0 and standard deviation = 1.
• General relationships: ±1 s = about 68.26%
±2 s = about 95.44%
±3 s = about 99.72%
68.26%
95.44%
99.72%
-5
-4
-3
-2
-1
0
1
2
3
4
5
Notes about z-scores
Are a way of determining the position of a single
score under the normal curve.
Measured in standard deviations relative to the
mean of the curve.
The Z-score can be used to determine an area
under the curve known as a probability.
z = (y - µ)
σ
• The standard normal distribution
is the normal distribution with µ = 0, σ = 1
For that distribution, z = (y - µ)/σ = (y - 0)/1 = y
i.e., original score = z-score
µ + zσ = 0 + z(1) = z
• Why is normal distribution so important?
If different studies take random samples and calculate a
statistic (e.g. sample mean) to estimate a parameter (e.g.
population mean), the collection of statistic values from
those studies usually has approximately a normal
distribution. (So?)
Notes about z-scores
Ex.
5000 tablets produced and assayed for content. The mean (µ ± σ) is 200 ± 10 mg
and the concentration is normally distributed.
Calculate the proportion of tablets that contain 180 mg or less.
68.26%
z = (y - µ)
σ
95.44%
99.72%
-5
-4
-3
-2
-1
-40
-30
-20
-10
160
170
180
190
0
5
z
1
2
3
4
0
10
20
30
40
y-µ (mg)
200
210
220
240
240
Y (mg)
z=
𝑦−µ
σ
180−200
10
=
= -2.00
68.26%
95.44%
99.72%
-5
-4
-3
-2
-1
0
1
2
3
4
5
probability distribution z-score table
(z probability table)
(http://www.bucknam.com/zprob.html)
z= - 2.00, probability below 0.023%
A sampling distribution lists the possible values
of a statistic (e.g., sample mean or sample
proportion) and their probabilities
How close is sample mean Ῡ to population mean µ?
To answer this, we must be able to answer,
“What is the probability distribution of the sample
mean?”
Sampling distribution of sample mean
•
y
is a variable, its value varying from sample to sample
about the population mean µ
• Standard deviation of sampling distribution of y is
called the standard error of y
• For random sampling, the sampling distribution of
y has mean µ and standard error
s
population standard deviation
sy 

n
sample size
Central Limit Theorem: For random sampling with
“large” n, the sampling distribution of the sample
mean is approximately a normal distribution
• Approximate normality applies no matter what the shape
of the population distribution.
• How “large” n needs to be depends on skew of population
distribution, but usually n ≥ 30 sufficient
5. Statistical Inference: Estimation
Goal: How can we use sample data to estimate values of
population parameters?
Point estimate: A single statistic value that is the “best
guess” for the parameter value
Interval estimate: An interval of numbers around the point
estimate, that has a fixed “confidence level” of
containing the parameter value. Called a confidence
interval.
(Based on sampling distribution of the point estimate)
Point Estimators – Most common to use
sample values
• Sample mean estimates population mean m
y

mˆ  y 
i
n
• Sample std. dev. estimates population std. dev. s
sˆ  s 
2
(
y

y
)
 i
n 1
• Sample proportion pˆ estimates population
proportion π
Confidence Intervals
• A confidence interval (CI) is an interval of numbers
believed to contain the parameter value.
Ex.
When public health practitioners use health statistics, sometimes they
are interested in the actual number of health events, but more often they
use the statistics to assess the true underlying risk of a health problem in
the community.
Statistical sampling theory is used to compute a confidence interval to
provide an estimate of the potential discrepancy between the true
population parameters and observed rates.
Understanding the potential size of that discrepancy can provide
information about how to interpret the observed statistic.
Confidence Intervals
• A confidence interval (CI) is an interval of numbers
believed to contain the parameter value.
• The probability the method produces an interval that contains the
parameter is called the confidence level. Most studies use a
confidence level close to 1, such as 0.95 or 0.99.
• Most CIs have the form
point estimate ± margin of error
with margin of error based on spread of sampling distribution of the
point estimator;
e.g., margin of error ±2(standard error) for 95% confidence.
Confidence Intervals
A 95% confidence interval for a percentage is
the range of scores within which the percentage
will be found if you went back and got a
different sample from the same population.
• The sampling distribution of a sample proportion for large
random samples is approximately normal
(Central Limit Theorem)
• So, with probability 0.95, sample proportion pˆ falls within
1.96 standard errors of population proportion π
• 0.95 probability that
z= 1.96 probability of 97.5% (or better 0.975)
pˆ falls between p  1.96s pˆ and p  1.96s pˆ
• Once sample selected, we’re 95% confident
pˆ  1.96s pˆ to pˆ  1.96s pˆ contains p
This is the CI for the population proportion π (almost)
Finding a CI in practice
• Complication: The true standard error
se =
s pˆ  s / n  p (1  p ) / n
𝑝 1−𝑝
𝑛
itself depends on the unknown parameter!
In practice, we estimate
s 
pˆ
p (1  p )
n
by se 
pˆ 1  pˆ 


n
and then find the 95% CI using the formula
pˆ  1.96(se) to pˆ  1.96(se)
(1-p)= q
Greater confidence requires wider CI
Greater sample size gives narrower CI
(quadruple n to halve width of CI)
Some comments about CIs
• Effects of n, confidence coefficient true for CIs for other
parameters also
• If we repeatedly took random samples of some fixed
size n and each time calculated a 95% CI, in the long run
about 95% of the CI’s would contain the population
proportion π.
• The probability that the CI does not contain π is called
the error probability, and is denoted by α.
• α = 1 – confidence coefficient
(1-)100%
90%
95%
99%

/2
.10
.05
.01
.050
.025
.005
z/2
1.645
1.96
2.58
Confidence Interval for the Mean
• In large random samples, the sample mean has
approximately a normal sampling distribution with
mean m and standard error
s
sy 
n
• Thus,
P(m  1.96s y  y  m  1.96s y )  .95
• We can be 95% confident that the sample mean lies
within 1.96 standard errors of the (unknown) population
mean
• Problem: Standard error is unknown (s is also a
parameter). It is estimated by replacing s with its point
estimate from the sample data:
s
se 
n
95% confidence interval for m :
s
y  1.96(se), which is y  1.96
n
This works ok for “large n,” because s then a good estimate of
σ (and CLT applies). But for small n, replacing σ by its
estimate s introduces extra error, and CI is not quite wide
enough unless we replace z-score by a slightly larger “t-score.”
The t distribution (Student’s t)
The t distribution is used instead of the normal distribution whenever the standard
deviation is estimated.
• Bell-shaped, symmetric about 0
• Standard deviation a bit larger than 1 (slightly thicker
tails than standard normal distribution, which has mean
= 0, standard deviation = 1)
• Precise shape depends on degrees of freedom (df). For
inference about mean,
df = n – 1
• Gets narrower and more closely resembles standard
normal distribution as df increases
(nearly identical when df > 30)
• CI for mean has margin of error t(se),
(instead of z(se) as in CI for proportion)
The t distribution (Student’s t)
Part of a t table
df
1
10
16
30
100
infinity
Confidence Level
90%
95%
98%
t.050
t.025
t.010
6.314 12.706 31.821
1.812
2.228
2.764
1.746
2.120
2.583
1.697
2.042
2.457
1.660
1.984
2.364
1.645
1.960
2.326
99%
t.005
63.657
3.169
2.921
2.750
2.626
2.576
df = ∞ corresponds to standard normal distribution
CI for a population mean
• For a random sample from a normal population
distribution, a 95% CI for µ is
y  t.025 (se), with se  s / n
where df = n - 1 for the t-score
• Normal population assumption ensures sampling
distribution has bell shape for any n.
Comments about CI for population mean µ
• The method is robust to violations of the assumption of a
normal population distribution
(But, be careful if sample data distribution is very highly
skewed, or if severe outliers. Look at the data.)
• Greater confidence requires wider CI
• Greater n produces narrower CI
• t methods developed by the statistician William Gosset of
Guinness Breweries, Dublin (1908)
Choosing the Sample Size
• Determine parameter of interest (population mean or
population proportion)
• Select a margin of error (M) and a confidence level
(determines z-score)
Proportion (to be “safe,” set p = 0.50):
Mean (need a guess for value of s):
 z 
n  p (1  p )  
M 
 z 
n s  
M 
2
2
2
• We’ve seen that n depends on confidence level (higher
confidence requires larger n) and the population
variability (more variability requires larger n)
• In practice, determining n not so easy, because (1) many
parameters to estimate, (2) resources may be limited
and we may need to compromise
• CI’s can be formed for any parameter.
Using CI Inference in Practice
• What is the variable of interest?
quantitative – inference about mean
categorical – inference about proportion
• Are conditions satisfied?
Randomization (why? Needed so sampling distribution and its
standard error are as advertised)
6. Statistical Inference:
Significance Tests
Goal: Use statistical methods to test hypotheses such as
“For treating anorexia, cognitive behavioral and family therapies have
same mean weight change as placebo” (no effect)
“Mental health tends to be better at higher levels of socioeconomic
status (SES)” (i.e., there is an effect)
“Spending money on other people has a more positive impact on
happiness than spending money on oneself.”
Hypotheses: For statistical inference, these are predictions
about a population expressed in terms of parameters (e.g.,
population means or proportions or correlations) for the
variables considered in a study
A significance test uses data to evaluate a hypothesis by
comparing sample point estimates of parameters to
values predicted by the hypothesis.
We answer a question such as, “If the hypothesis were true,
would it be unlikely to get data such as we obtained?”
Five Parts of a Significance Test
• Assumptions about type of data (quantitative, categorical),
sampling method (random), population distribution (e.g.,
normal, binary), sample size (large enough?)
• Hypotheses:
Null hypothesis (H0): A statement that parameter(s) take
specific value(s) (Usually: “no effect”)
Alternative hypothesis (Ha): states that parameter value(s)
falls in some alternative range of values
(an “effect”)
•
Test Statistic: Compares data to what null hypotesis H0
predicts, often by finding the number of standard errors
between sample point estimate and H0 value of
parameter
• P-value (P): A probability measure of evidence about H0.
The probability (under presumption that H0 true) the test
statistic equals observed value or value even more
extreme in direction predicted by Ha.
– The smaller the P-value, the stronger the evidence
against H0.
• Conclusion:
– If no decision needed, report and interpret P-value
– If decision needed, select a cutoff point (such as 0.05
or 0.01) and reject H0 if P-value ≤ that value
– The most widely accepted cutoff point is 0.05, and the
test is said to be “significant at the .05 level” if the Pvalue ≤ 0.05.
– If the P-value is not sufficiently small, we fail to reject H0
(then, H0 is not necessarily true, but it is plausible)
– Process is analogous to American judicial system
• H0: Defendant is innocent
• Ha: Defendant is guilty
Fine prima parte
Significance Test for Mean
• Assumptions: Randomization, quantitative variable, normal
population distribution (robustness?)
• Null Hypothesis: H0: µ = µ0 where µ0 is particular value for
population mean
(typically “no effect” or “no change” from a standard)
• Alternative Hypothesis: Ha: µ  µ0
2-sided alternative includes both > and < H0 value
• Test Statistic: The number of standard errors that the
sample mean falls from the H0 value
y  m0
t
where se  s / n
se
When H0 is true, the sampling distribution of the t test
statistic is the t distribution with df = n - 1.
• P-value: Under presumption that H0 true, probability the t
test statistic equals observed value or even more extreme
(i.e., larger in absolute value), providing stronger evidence
against H0
– This is a two-tail probability, for the two-sided Ha
• Conclusion: Report and interpret P-value. If needed, make
decision about H0
Making a decision:
The α-level is a fixed number, also called the significance level,
such that
if P-value ≤ α, we “reject H0”
If P-value > α, we “do not reject H0”
Note: We say “Do not reject H0” rather than “Accept H0”
because H0 value is only one of many plausible values.
A high significance level means there is a large chance
that the experiment proves something that is not true.
A very small significance level assures the statistician that
there is little room to doubt the results.
Effect of sample size on tests
• With large n (say, n > 30), assumption of normal population
distribution not important because of Central Limit Theorem.
• For small n, the two-sided t test is robust against violations of that
assumption. However, one-sided test is not robust.
• For a given observed sample mean and standard deviation, the
larger the sample size n, the larger the test statistic (because se in
denominator is smaller) and the smaller the P-value. (i.e., we
have more evidence with more data)
• We’re more likely to reject a false H0 when we have a larger
sample size (the test then has more “power”)
• With large n, “statistical significance” not the same as “practical
significance.”
Significance Test for a Proportion π
• Assumptions:
– Categorical variable
– Randomization
– Large sample (but two-sided ok for nearly all n)
• Hypotheses:
– Null hypothesis: H0: p  p0
– Alternative hypothesis: Ha: p  p0 (2-sided)
– Ha: p > p0
Ha: p < p0 (1-sided)
– Set up hypotheses before getting the data
• Test statistic:
Note
pˆ  p 0
pˆ  p 0
z

s
p 0 (1  p 0 ) / n
pˆ
s pˆ  se0  p 0 (1  p 0 ) / n , not se  pˆ (1  pˆ ) / n as in a CI
As in test for mean, test statistic has form
(estimate of parameter – H0 value)/(standard error)
= no. of standard errors the estimate falls from H0 value
• P-value:
Ha: p  p0 P = 2-tail prob. from standard normal dist.
Ha: p > p0 P = right-tail prob. from standard normal dist.
Ha: p < p0 P = left-tail prob. from standard normal dist.
• Conclusion: As in test for mean (e.g., reject H0 if P-value ≤ α)
Decisions in Tests
 -level (significance level): Pre-specified “hurdle” for
which one rejects H0 if the P-value falls below it
(Typically 0.05 or 0.01)
P-Value
.05
> .05
H0 Conclusion
Reject
Do not Reject
Ha Conclusion
Accept
Do not Accept
• Rejection Region: Values of the test statistic for which we
reject the null hypothesis
 For 2-sided tests with  = 0.05, we reject H0 if |z| 1.96
Error Types
• Type I Error: Reject H0 when it is true
• Type II Error: Do not reject H0 when it is false
Test Result –
Reject H0
Don’t Reject
H0
Reality
H0 True
Type I Error
Correct
H0 False
Correct
Type II Error
P(Type I error)
• Suppose -level = 0.05. Then, P(Type I error) = P(reject
null, given it is true) = P(|z| > 1.96) = 0.05
• i.e., the -level is the P(Type I error).
• Since we “give benefit of doubt to null” in doing test, it’s
traditional to take  small, usually 0.05 but 0.01 to be very
cautious not to reject null when it may be true.
• As in CIs, don’t make  too small, since as  goes down, β =
P(Type II error) goes up
(Think of analogy with courtroom trial)
• Better to report P-value than merely whether reject H0
P(Type II error)
• P(Type II error) = b depends on the true value of the
parameter (from the range of values in Ha ).
• The farther the true parameter value falls from the null
value, the easier it is to reject null, and P(Type II error) goes
down.
• Power of test = 1 -α = P(reject null, given it is false)
• In practice, you want a large enough n for your study so
that P(Type II error) is small for the size of effect you
expect.
Practical Applications of Statistics

researchers use a test of significance to determine
whether to reject or fail to reject the null
hypothesis
…involves pre-selecting a level of probability, “α”
(e.g., α = .05) that serves as the criterion to
determine whether to reject or fail to reject the
null hypothesis
Steps in using inferential statistics…
1. select the test of significance
2. determine whether significance test will be twotailed or one tailed
3. select α (alpha), the probability level
4. compute the test of significance
5. consult table to determine the significance of the
results
Tests of significance...
• statistical formulas that enable the researcher to
determine if there was a real difference between
the sample means
different tests of significance account for different
factors including: the scale of measurement
represented by the data; method of participant
selection, number of groups being compared, and, the
number of independent variables
…the researcher must first decide whether a
parametric or nonparametric test must be
selected

parametric test...
•
assumes that the variable measured is normally
distributed in the population
•
the selection of participants is independent
•
the variances of the population comparison
groups are equal
used when the data represent a interval or ratio scale

nonparametric test...
makes no assumption about the distribution of
the variable in the population, that is, the shape
of the distribution
• used when the data represent a nominal or
ordinal scale, when a parametric assumption has
been greatly violated, or when the nature of the
distribution is not known
•
•
usually requires a larger sample size to reach the
same level of significance as a parametric test
• The most common tests of significance…
t-test
z-test
ANOVA
Chi Square

t-test...
…used to determine whether two means are
significantly different at a selected probability
level
…adjusts for the fact that the distribution of scores for
small samples becomes increasingly different from
the normal distribution as sample sizes become
increasingly smaller
the strategy of the t-test is to compare
the actual mean difference observed
to the difference expected by chance

t-test...
forms a ratio where the numerator is the difference
between the sample means and the denominator is
the chance difference that would be expected if the
null hypothesis were true
after the numerator is divided by the denominator,
the resulting t value is compared to the appropriate
t table value, depending on the probability level and
the degrees of freedom
if the t value is equal to or greater than the table
value, then the null hypothesis is rejected because
the difference is greater than would be expected
due to chance (t stat > t tab, H0 rejected)

t-test...
there are two types of t-tests:
the t-test for independent samples (randomly formed)
the t-test for nonindependent samples (nonrandomly
formed, e.g., matching, performance on a pre-/posttest, different treatments)

t-test...
Ex. t-test

Z-test...
Ex. Z-test

ANOVA (Analysis of Variance)
used to determine whether two or more
means are significantly different at a selected
probability level
avoids the need to compute duplicate t-tests to
compare groups (more than 2)
the strategy of ANOVA is that total variation, or variance,
can be divided into two sources:
treatment variance
“between groups,” variance caused by the treatment groups
error variance
“within groups” variance

ANOVA (Analysis of Variance)
forms a ratio, the F ratio, with the treatment
variance as the numerator (between group
variance) and error variance as the
denominator (within group variance)
…the assumption is that randomly formed groups
of participants are chosen and are essentially
the same at the beginning of a study on a
measure of the dependent variable
…at the study’s end, the question is whether the
variance between the groups differs from the
error variance by more than what would be
expected by chance
•
if the treatment variance is sufficiently larger
than the error variance, a significant F ratio
results, that is, the null hypothesis is rejected
and it is concluded that the treatment had a
significant effect on the dependent variable
•
if the treatment variance is not sufficiently
larger than the error variance, an insignificant
F ratio results, that is, the null hypothesis is
accepted and it is concluded that the
treatment had no significant effect on the
dependent variable
when the F ratio is significant and more than
two means are involved, researchers use
multiple comparison procedures (e.g., Scheffé
test, Tukey’s HSD test, Duncan’s multiple range
test)

ANOVA (Analysis of Variance)
Ex. ANOVA
One-way
Two-way

Chi Square (Χ2) Pearson’s test
a nonparametric test of significance appropriate for
nominal or ordinal data that can be converted to
frequencies
It can be applied to interval or ratio data that have been
categorized into a small number of groups.
It assumes that the observations are randomly sampled from
the population.
All observations are independent (an individual can appear
only once in a table and there are no overlapping
categories).
It does not make any assumptions about the shape of the
distribution nor about the homogeneity of variances.

Chi Square (Χ2)...
compares the proportions actually observed (O) to the
proportions expected (E) to see if they are
significantly different
…the chi square value increases as the difference
between observed and expected frequencies
increases
One- and two- tailed tests of significance...
• tests of significance that indicate the direction in
which a difference may occur
…the word “tail” indicates the area of rejection
beneath the normal curve
• Ho : The two variables are independent
• Ha : The two variables are associated

Chi Square (Χ2) calculations
• Contrasts observed frequencies in each cell of a
contingency table with expected frequencies.
• The expected frequencies represent the number
of cases that would be found in each cell if the
null hypothesis were true ( i.e. the nominal
variables are unrelated).
• Expected frequency of two unrelated events is
product of the row and column frequency divided
by number of cases.
Fe= Fr Fc / N
 ( Fo  Fe ) 
  

Fe


2
2
Determine Degrees of Freedom
• df = (R-1)(C-1)
Compare computed test statistic
against a tabled/critical value
• The computed value of the Pearson chisquare statistic is compared with the critical
value to determine if the computed value is
improbable
• The critical tabled values are based on
sampling distributions of the Pearson chisquare statistic
• If calculated 2 is greater than 2 table
value, reject Ho
• A=B
no difference between means; the direction can be
positive or negative
direction can be in either tail of the normal curve
called a “two-tailed” test
divides the α level between the two tails of the
normal curve
• A > B or A < B
there is a difference between means; the
direction is either positive or negative
called a “one-tailed” test
the α level is found in one tail of the normal curve
Ex. Chi Square (Χ2)
3. Bivariate description
• Usually we want to study associations between two or
more variables (e.g., how does number of close friends
depend on gender, income, education, age, working status,
rural/urban, religiosity…)
• Response variable: the outcome variable
• Explanatory variable(s): defines groups to compare
Ex.: number of close friends is a response variable, while
gender, income, … are explanatory variables
Response var. also called “dependent variable”
Explanatory var. also called “independent variable”
Summarizing associations:
• Categorical var’s: show data using contingency tables
• Quantitative var’s: show data using scatterplots
• Mixture of categorical var. and quantitative var. (e.g.,
number of close friends and gender) can give numerical
summaries (mean, standard deviation) or side-by-side box
plots for the groups
Contingency Tables
• Cross classifications of categorical variables in which
rows (typically) represent categories of explanatory
variable and columns represent categories of response
variable.
• Counts in “cells” of the table give the numbers of
individuals at the corresponding combination of levels of
the two variables
Another Example Heparin Lock
Placement
Time:
1 = 72 hrs
2 = 96 hrs
Complication Incidence * Heparin Lock Placement Time Group Crosstabulation
Complication
Incidence
Had Compilca
Had NO Compilca
Total
Count
Expected Count
% within Heparin Lock
Placement Time Group
Count
Expected Count
% within Heparin Lock
Placement Time Group
Count
Expected Count
% within Heparin Lock
Placement Time Group
Heparin Lock
Placement Time Group
1
2
9
11
10.0
10.0
Total
20
20.0
18.0%
22.0%
20.0%
41
40.0
39
40.0
80
80.0
82.0%
78.0%
80.0%
50
50.0
50
50.0
100
100.0
100.0%
100.0%
100.0%
147
Hypotheses in Heparin Lock Placement
• Ho: There is no association between
complication incidence and length of heparin
lock placement. (The variables are
independent).
• Ha: There is an association between
complication incidence and length of heparin
lock placement. (The variables are related).
148
More of SPSS Output
149