Univariate, bivariate analysis, hypothesis testing, chi square

Introduction to Data Analysis
•Data Measurement
•Measurement of the data is the first step in the process that ultimately
guides the final analysis.
•Consideration of sampling, controls, errors (random and systematic)
and the required precision all influence the final analysis.
•Validation: Instruments and methods used to measure the data must
be validated for accuracy.
•Precision and accuracy…Determination of error
•Social vs. Physical Sciences
1

•Types of data
•Univariate/Multivariate
•Univariate: When we use one variable to describe a person, place,
or thing. (e.g. Heights of individuals)
•Multivariate: When we use two or more variables to measure a
person, place or thing. Variables may or may not be dependent on
each other.
(Bivariate e.g. name and marks, Multivariate: name, caste and marks)
•Cross-sectional data/Time-ordered data (business, social sciences)
•Cross-Sectional: Measurements taken at one time period
•E.g. (caste and per capita incomes)
•Time-Ordered: Measurements taken over time in chronological
sequence. e.g. years and per capita income
The type of data will dictate (in part) the appropriate data-analysis method.
2

•Measurement Scales
•Nominal or Categorical Scale (e.g. fair, brown and black)
•Classification of people, places, or things into categories (e.g. age
ranges, colors, etc.).
•Classifications must be mutually exclusive (every element should
belong to one category with no ambiguity).
•Weakest of the four scales. No category is greater than or less
(better or worse) than the others. They are just different.
•Ordinal or Ranking Scale
•Classification of people, places, or things into a ranking such that
the data is arranged into a meaningful order (e.g. poor, fair, good,
excellent).
•Qualitative classification only
3

•Measurement Scales (business, social sciences)
•Interval Scale
•Data classified by ranking.
•Quantitative classification (time, temperature, etc).
•Zero point of scale is arbitrary (differences are meaningful).
•Ratio Scale
•Data classified as the ratio of two numbers.
•Quantitative classification (height, weight, distance, etc).
•Zero point of scale is real
•(data can be added, subtracted, multiplied, and divided).
4

Univariate Analysis/Descriptive Statistics
• Descriptive Statistics
– The Range
– Min/Max
– Average
– Median
– Mode
– Variance
– Standard Deviation
– Histograms and Normal Distributions
5

Univariate Analysis/Histograms
• Distributions
– Descriptive statistics are easier to interpret when
graphically illustrated.
– However, charting each data element can lead to very
busy and confusing charts that do not help interpret the
data.
– Grouping the data elements into categories and
charting the frequency within these categories yields a
graphical illustration of how the data is distributed
throughout its range.
6

0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
X-axis labels
DataValues
With just a few columns this chart is difficult to interpret. It tells you very little
about the data set. Even finding the Min and Max can be difficult.
The data can be presented such that more statistical parameters can be
estimated from the chart (average, standard deviation).
7

• Frequency Table
– The first step is to decide on the categories and group
the data appropriately.
(45, 49, 50, 53, 60, 62, 63, 65, 66, 67, 69, 71, 73, 74, 74,
78, 81, 85, 87, 100)
Category Labels Frequency
0-50 3
51-60 2
61-70 6
71-80 5
81-90 3
>90 1
8

• Histogram
– A histogram is simply a column chart of the frequency
table.
Category Labels Frequency
0-50 3
51-60 2
61-70 6
71-80 5
81-90 3
>90 1
0
1
2
3
4
5
6
7
0-50 51-60 61-70 71-80 81-90 >90
Scores
Frequency
9

• Histogram
0
1
2
3
4
5
6
7
0-50 51-60 61-70 71-80 81-90 >90
Scores
Frequency
Average (68.6)
and Median (68)
Mode (74)
-1SD
+1SD
10

0
0.02
0.04
0.06
0.08
0.1
0.12
25 45 65 85 105 125 145 165
Univariate Analysis/Normal Distributions
• Distributions that can be described mathematically
as Gaussian are also called Normal
• The Bell curve
– Symmetrical
– Mean ≈ Median
Mean, Median, Mode
11

Univariate Analysis/Skewed Distributions
• When data are skewed, the
mean and SD can be
misleading
• Skewness
sk= 3(mean-median)/SD
If sk>|1| then distribution is
non-symetrical
• Negatively skewed
– Mean<Median
– Sk is negative
• Positively Skewed
– Mean>Median
– Sk is positive
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 20 40 60 80 100 120 140 160
0
0.02
0.04
0.06
0.08
0.1
0.12
25 45 65 85 105 125 145 165 185 205 225
12

Central Limit Theorem
• Regardless of the shape of a distribution, the distribution
of the sample mean based on samples of size N
approaches a normal curve as N increases.
– N must be less than the entire sample
N=10
13

• The Range
– Difference between minimum and maximum
values in a data set
– Larger range usually (but not always)
indicates a large spread or deviation in the
values of the data set.
(73, 66, 69, 67, 49, 60, 81, 71, 78, 62, 53, 87,
74, 65, 74, 50, 85, 45, 63, 100)
14

• The Average (Mean)
– Sum of all values divided by the number of values in the data set.
– One measure of central location in the data set.
Average =
Average=(73+66+69+67+49+60+81+71+78+62+53+87+74+65+74
+50+85+45+63+100)/20 = 68.6
Excel function: AVERAGE()
∑=
N
i
im
N 1
1
15

0 2.5 7.5 10
4.8
0 2.5 7.5 10
4.8
The data may or
may not be
symmetrical around
its average value
16

• The Median
– The middle value in a sorted data set. Half the values
are greater and half are less than the median.
– Another measure of central location in the data set.
(45, 49, 50, 53, 60, 62, 63, 65, 66, 67, 69, 71, 73, 74, 74,
78, 81, 85, 87, 100)
Median: 68
(1, 2, 4, 7, 8, 9, 9)
– Excel function: MEDIAN()
17

• The Median
– May or may not be close to the mean.
– Combination of mean and median are used to define
the skewness of a distribution.
0 2.5 7.5 10
6.25
18

• The Mode
– Most frequently occurring value.
– Another measure of central location in the data set.
– (45, 49, 50, 53, 60, 62, 63, 65, 66, 67, 69, 71, 73, 74,
74, 78, 81, 85, 87, 100)
– Mode: 74
– Generally not all that meaningful unless a larger
percentage of the values are the same number.
19

• Variance
– One measure of dispersion (deviation from the mean) of a data
set. The larger the variance, the greater is the average deviation
of each datum from the average value.
=
−∑=
m
mm
N
N
i
i
2
1
)(
1
Variance =
Average value of the data set
Variance = [(45 – 68.6)2
+ (49 – 68.6)2
+ (50 – 68.6)2
+ (53 – 68.6)2
+ …]/20 = 181
Excel Functions: VARP(), VAR()
20

• Standard Deviation
– Square root of the variance. Can be thought of as the
average deviation from the mean of a data set.
– The magnitude of the number is more in line with the
values in the data set.
Standard Deviation = ([(45 – 68.6)2
+ (49 – 68.6)2
+ (50 – 68.6)2
+ (53 – 68.6)2
+
…]/20)1/2
= 13.5
Excel Functions: STDEVP(), STDEV()
21

Bivariate Analysis
Cross-tabulation and chi-square
22

So far the statistical methods we
have used only permit us to:
• Look at the frequency in which certain
numbers or categories occur.
• Look at measures of central tendency such
as means, modes, and medians for one
variable.
• Look at measures of dispersion such as
standard deviation and z scores for one
interval or ratio level variable.
23

Bivariate analysis allows us to:
• Look at associations/relationships among
two variables.
• Look at measures of the strength of the
relationship between two variables.
• Test hypotheses about relationships between
two nominal or ordinal level variables.
24

For example, what does this table tell us about
opinions on welfare by gender?
Support cutting
welfare benefits
for immigrants
Male Female
Yes 15 5
No 10 20
Total 25 25
25

Are frequencies sufficient to
allow us to make comparisons
about groups?
What other information do we
need?
26

Is this table more helpful?
Benefits for
Immigrants
Males Female
Yes 15 (60%) 5 (20%)
No 10 (40%) 20 (80%)
Total 25 (100%) 25 (100%)
27

How would you write a sentence
or two to describe what is in this
table?
28

Rules for cross-tabulation
• Calculate either column or row percents.
• Calculations are the number of frequencies
in a cell of a table divided by the total
number of frequencies in that column or
row, for example 20/25 = 80.0%
• All percentages in a column or row should
total 100%.
29

Let’s look at another example –
social work degrees by gender
Social Work
Degree
Male Female
BA 20 (33.3%) 20 ( %)
MSW 30 ( ) 70 (70.0%)
Ph.D. 10 (16.7%) 10 (10.0%)
60 (100.0%) 100 (100.0%
30

Questions:
What group had the largest percentage of
Ph.Ds?
What are the ways in which you could
find the missing numbers?
Is it obvious why you would use
percentages to make comparisons among
two or more groups? 31

In the following table, were people with drug,
alcohol, or a combination of both most likely
to be referred for individual treatment?
Services Alcohol Drugs Both
Individual
Treatment
10 (25%) 30 (60%) 5 (50%)
Group
Treatment
10 (25%) 10 (20%) 2 (20%)
AA 20 (50%) 10 (20%) 3 (30%)
Total 40 (100%) 50 (100%) 10 (100%)
32

Use the same table to answer the
following question:
How much more likely are
people with alcohol problems
alone to be referred to AA than
people with drug problems or a
combination of drug and alcohol
problems? 33

We use cross-tabulation when:
• We want to look at relationships among two
or three variables.
• We want a descriptive statistical measure to
tell us whether differences among groups
are large enough to indicate some sort of
relationship among variables.
34

Cross-tabs are not sufficient to:
• Tell us the strength or actually size of the relationships
among two or three variables.
• Test a hypothesis about the relationship between two or
three variables.
• Tell us the direction of the relationship among two or more
variables.
• Look at relationships between one nominal or ordinal
variable and one ratio or interval variable unless the range
of possible values for the ratio or interval variable is small.
What do you think a table with a large number of ratio
values would look like?
35

We can use cross-tabs to visually
assess whether independent and
dependent variables might be
related. In addition, we also use
cross-tabs to find out if
demographic variables such as
gender and ethnicity are related
to the second variable.
36

For example, gender may
determine if someone votes
Democratic or Republican or if
income is high, medium, or low.
Ethnicity might be related to
where someone lives or attitudes
about whether undocumented
workers should receive driver’s
licenses. 37

Because we use tables in these ways, we can
set up some decision rules about how to use
tables.
• Independent variables should be column variables.
• If you are not looking at independent and
dependent variable relationships, use the variable
that can logically be said to influence the other as
your column variable.
• Using this rule, always calculate column
percentages rather than row percentages.
• Use the column percentages to interpret your
results.
38

For example,
• If we were looking at the relationship between gender and
income, gender would be the column variable and income
would be the row variable. Logically gender can determine
income. Income does not determine your gender.
• If we were looking at the relationship between ethnicity
and location of a person’s home, ethnicity would be the
column variable.
• However, if we were looking at the relationship between
gender and ethnicity, one does not influence the other.
Either variable could be the column variable.
39

SPSS will allow you to choose a
column variable and row variable
and whether or not your table
will include column or row
percents.
40

You must use an additional statistic, chi-
square, if you want to:
• Test a hypothesis about two variables.
• Look at the strength of the relationship between an
independent and dependent variable.
• Determine whether the relationship between the
two variables is large enough to rule out random
chance or sampling error as reasons that there
appears to be a relationship between the two
variables.
41

Chi-square is simply an extension of a
cross-tabulation that gives you more
information about the relationship.
However, it provides no information
about the direction of the relationship
(positive or negative) between the two
variables.
42

Let’s use the following table to
test a hypothesis:
Education
Income High Low Total
High (Above
$40,000)
40 50
Low ($39,999
or less)
50
Total 50 50 100
43

I have not filled in all of the information
because we need to talk about two concepts
before we start calculations:
• Degrees of Freedom: In any table, there are
a limited number of choices for the values
in each cell.
• Marginals: Total frequencies in columns
and rows.
44

Let’s look at the number of choices
we have in the previous table:
Education
High (Above
$40,000)
40 50
Low ($39,999
or less)
50
Total 50 50 100
45

So the table becomes:
Education
High (Above
$40,000)
40 10 50
Low ($39,999
or less)
10 40 50
Total 50 50 100
46

The rules for determining degrees of freedom
in cross-tabulations or contingency tables:
• In any two by two tables (two columns, two
rows, excluding marginals) DF = 1.
• For all other tables, calculate DF as:
(c -1 ) * (r-1) where c = columns and r =
rows.
( So for a table with 3 columns and 4 rows,
DF = ____. )
47

Importance of Degrees of Freedom
• You will see degrees of freedom on your SPSS
print out.
• Most types of inferential statistics use DF in
calculations.
• In chi-square, we need to know DF if we are
calculating chi-square by hand. You must use the
value of the chi-square and DF to determine if the
chi-square value is large enough to be statistically
significant (consult chi-square table in most
statistics books).
48

Hypothesis Testing
• Goal: Make statement(s) regarding unknown population
parameter values based on sample data
• Elements of a hypothesis test:
– Null hypothesis - Statement regarding the value(s) of unknown
parameter(s). Typically will imply no association between
explanatory and response variables in our applications (will
always contain an equality)
– Alternative hypothesis - Statement contradictory to the null
hypothesis (will always contain an inequality)
– Test statistic - Quantity based on sample data and null
hypothesis used to test between null and alternative hypotheses
– Rejection region - Values of the test statistic for which we
reject the null in favor of the alternative hypothesis
49

Hypothesis Testing
Test Result –
True State
H0 True H0 False
H0 True Correct
Decision
Type I Error
H0 False Type II Error Correct
Decision
)()( ErrorIITypePErrorITypeP == βα
• Goal: Keep α, β reasonably small 50

Example - Efficacy Test for New drug
• Drug company has new drug, wishes to compare it
with current standard treatment
• Federal regulators tell company that they must
demonstrate that new drug is better than current
treatment to receive approval
• Firm runs clinical trial where some patients
receive new drug, and others receive standard
treatment
• Numeric response of therapeutic effect is obtained
(higher scores are better).
• Parameter of interest: µNew - µStd
51

• Null hypothesis - New drug is no better than standard trt
( )00:0 =−≤− StdNewStdNewH µµµµ
• Alternative hypothesis - New drug is better than standard trt
0: >− StdNewAH µµ
• Experimental (Sample) data:
StdNew
StdNew
StdNew
nn
ss
yy
52

Sampling Distribution of Difference in Means
• In large samples, the difference in two sample means is
approximately normally distributed: N= Normal
distribution, with a mean and SD








+−−
2
2
2
1
2
1
2121 ,~
nn
NYY
σσ
µµ
• Under the null hypothesis, µ1-µ2=0 and:
)1,0(~
2
2
2
1
2
1
21
N
nn
YY
Z
σσ
+
−
=
53

• Type I error - Concluding that the new drug is better than the
standard (HA) when in fact it is no better (H0). Ineffective drug is
deemed better.
– Traditionally α = P(Type I error) = 0.05
• Type II error - Failing to conclude that the new drug is better (HA)
when in fact it is. Effective drug is deemed to be no better.
– Traditionally a clinically important difference (∆) is assigned
and sample sizes chosen so that:
β = P(Type II error | µ1-µ2 = ∆) ≤ 0.20
54

Elements of a Hypothesis Test
• Test Statistic - Difference between the Sample means,
scaled to number of standard deviations (standard errors)
from the null difference of 0 for the Population means:
2
2
2
1
2
1
21
:..
n
s
n
s
yy
zST obs
+
−
=
• Rejection Region - Set of values of the test statistic that are consistent
with HA, such that the probability it falls in this region when H0 is true is α
(we will always set α=0.05)
645.105.0:.. =⇒=≥ αα α zzzRR obs
55

P-value (aka Observed Significance Level)
• P-value - Measure of the strength of evidence the sample
data provides against the null hypothesis:
P(Evidence This strong or stronger against H0 | H0 is true)
)(: obszZPpvalP ≥=−
56

Large-Sample Test H0:µ1-µ2=0 vs H0:µ1-µ2>0
• H0: µ1-µ2 = 0 (No difference in population means
• HA: µ1-µ2 > 0 (Population Mean 1 > Pop Mean 2)
ty_value][probabiliobs
obs
2
2
2
1
2
1
21
obs
)zZ(P:valueP
zz:.R.R
n
s
n
s
yy
z:.S.T
Region][Rejection
Statistic][Test
=≥−•
≥=•
+
−
==•
α
• Conclusion - Reject H0 if test statistic falls in rejection region, or
equivalently the P-value is ≤ α
57

Example - Botox for Cervical Dystonia
• Patients - Individuals suffering from cervical dystonia
• Response - Tsui score of severity of cervical dystonia
(higher scores are more severe) at week 8 of Tx
• Research (alternative) hypothesis - Botox A decreases
mean Tsui score more than placebo
• Groups - Placebo (Group 1) and Botox A (Group 2)
• Experimental (Sample) Results:
354.37.7
336.31.10
222
111
===
===
nsy
nsy
Source: Wissel, et al (2001)
58

Example - Botox for Cervical Dystonia
0024.)82.2(:
645.1:..
82.2
85.0
4.2
35
)4.3(
33
)6.3(
7.71.10
:..
0:
0:
05.
22
21
210
=≥−•
==≥•
==
+
−
=•
>−•
=−•
ZPvalP
zzzRR
zST
H
H
obs
obs
A
α
µµ
µµ
Test whether Botox A produces lower mean Tsui scores than
placebo (α = 0.05)
Conclusion: Botox A produces lower mean Tsui scores than placebo
(since 2.82 > 1.645 and P-value < 0.05)
There is only 0.24%
chance that it is by
chance. Hence
Botox is better.
59

2-Sided Tests
• Many studies don’t assume a direction wrt the
difference µ1-µ2
• H0: µ1-µ2 = 0 HA: µ1-µ2 ≠ 0
• Test statistic is the same as before
• Decision Rule:
– Conclude µ1-µ2> 0 if zobs ≥ zα/2 (α=0.05 ⇒ zα/2=1.96)
– Conclude µ1-µ2< 0 if zobs ≥ -zα/2 (α=0.05 ⇒ -zα/2= -1.96)
– Do not reject µ1-µ2 = 0 if -zα/2 ≤ zobs ≤ zα/2
• P-value: 2P(Z≥ |zobs|)
60

Power of a Test
• Power - Probability a test rejects H0(depends on µ1- µ2)
– H0 True: Power = P(Type I error) = α
– H0 False: Power = 1-P(Type II error) = 1-β
· Example:
· H0: µ1- µ2 = 0 HA: µ1- µ2 > 0
• σ1
2
= σ2
2
= 25 n1 = n2 = 25
· Decision Rule: Reject H0 (at α=0.05 significance level) if:
326.2645.1
2
21
21
2
2
2
1
2
1
21
≥−⇒≥
−
=
+
−
= yy
yy
nn
yy
zobs
σσ
1.414* 1.645= 2.326
61

Power of a Test
• Now suppose in reality that µ1-µ2 = 3.0 (HA is true)
• Power now refers to the probability we (correctly)
reject the null hypothesis. Note that the sampling
distribution of the difference in sample means is
approximately normal, with mean 3.0 and standard
deviation (standard error) 1.414.
• Decision Rule (from last slide): Conclude population
means differ if the sample mean for group 1 is at least
2.326 higher than the sample mean for group 2
• Power for this case can be computed as:
)414.10.2,3(~)326.2( 2121 =−≥− NYYYYP
62

Power of a Test
• All else being equal:
• As sample sizes increase, power increases
• As population variances decrease, power
increases
• As the true mean difference increases, power
increases
63

Power of a Test
Distribution (H0) Distribution (HA)
64

Power of a Test
Power Curves for group sample sizes of 25,50,75,100 and varying true
values µ1-µ2 with σ1=σ2=5.
• For given µ1-µ2 , power increases with sample size
• For given sample size, power increases with µ1-µ2 65

Steps in testing a hypothesis:
• State the research hypothesis
• State the null hypothesis
• Choose a level of statistical significance
(alpha level)
• Select and compute the test statistic
• Make a decision regarding whether to
accept or reject the null hypothesis.
66

Calculating Chi-Square
• Formula is [0 - E]2
E
Where 0 is the observed value in a cell
E is the expected value in the same
cell we would see if there was no
association
67

First steps
Alternative hypothesis is: There is a relationship
between income level and education for
respondents in a survey of BA students.
Null hypothesis is: There is no relationship between
income level and education for respondents in a
survey of BA students
Confidence level set at 0.05
68

Rules for determining whether the chi-square
statistic and probability are large enough to verify a
relationship.
• For hand calculations, use the degree(s) of
freedom and the confidence level you set to check
the Chi-square table found in most statistics
books. For the chi-square to be statistically
significant, it must be the same size or larger than
the number in the table.
• On an SPSS print out, the p. or significance value
must be the same size or smaller than your
significance level.
69

The formula for expected values are
E = R*C
Education
High (Above
$40,000)
25 25 50
Low ($39,999
or less)
25 25 50
Total 50 50 100
70

Go back to our first table
Education
High (Above
$40,000)
40 10 50
Low ($39,999
or less)
10 40 50
Total 50 50 100
71

Chi-square calculation is
Expected
Values Chi-square
Cell 1 50 * 50/100= 25 (40-25)2
/25= 9
Cell 2 50*50/100= 25 (10-25)2
/25= 9
Cell 3 50 * 50/100= 25 (10-25)2
/25= 9
Cell 4 50*50/100= 25 (40-25)2
/25= 9
36
At 0.05, 1 = df, chi-square must be larger
than 3.84 to be statistically significant
72

Let’s calculate another chi-square- service
receipt by location of residence
Service Urban Rural Total
Yes 20 40 60
No 30 10 40
Total 50 50 100
74

For this table,
• DF = 1
• Alternative hypothesis:
Receiving service is associated with
location of residence.
Null hypothesis:
There is no association between receiving
service and location of residence.
75

Calculations for chi-square are
Expected
Values Chi-square
Cell 1 50 * 60/100= 30 (20-30)2
/30= 3.33
Cell 2 50*40/100= 20 (30-20)2
/20= 5.00
Cell 3 50*60/100= 30 (40-30)2
/30= 3.33
Cell 4 50*40/100= 20 (10-20)2
/20= 5.00
16.67
At 1 DF at 0.01 chi-square must be greater than 6.64.
Do we accept or reject the null hypothesis? 76

Running chi-square in SPSS
• Select descriptive statistics
• Select cross-tabulation
• Highlight your independent variable and click on the arrow.
• Highlight your dependent variable and click on the arrow.
• Select Cells
• Choose column percents
• Click continue
• Select statistics
• Select chi-square
• Click continue
• Click ok
77

Univariate, bivariate analysis, hypothesis testing, chi square

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