data analysis (1) qual qual
DESCRIPTION
Data AnalysisTRANSCRIPT
Data Analysis (1/3)
Typical steps for a Statistical study
1. Define the goals
2. Collect the data
3. Organize the data
4. Present the data
5. Describe the data
6. Analyze the data
7. Interpret results 3
How this
lecture fits up
to this
point...
Is there a correlation/association/relationship/interaction between the variables?
Dependence:
o Relationship exists between variables
o Change in one variable is accompanied by change in the other variable (these two things seem to happen at the same time).
Independence: o No relationship exists
between variables
o Change in one variable is NOT accompanied by change in the other variable.
Positive relationship Negative relationship
No relationship
As x increases, Y increases
As x increases, Y decreases
As x increases, Y doesn’t change
Which statement makes more sense:
1) The age of a bus can influence the maintenance cost.
2) The maintenance cost can influence the age of the bus.
X-axis is the INDEPENDENT
Variable (Age) which influences the
dependent variable.
Y axis is the
DEPENDENT
variable (Cost)
which is influenced.
To study a relationship, one variable is called the DEPENDENT variable and the other is called the INDEPENDENT variable.
The dependent variable may be explained/predicted/influenced/
understood by the independent variable.
Data Analysis and Tools
Chi-Square Chi-Square
Numerical
Categorical Numerical
Categorical
Independent Variable (X) De
pe
nd
ent
Var
iab
le (
Y)
The data and the type of research question you want answered will determine the most appropriate analytical
procedure to select.
Let’s go!
There are two contingency table tests:
(1) Two-way table contingency test (also called “test of independence”)
(2) One-way table contingency test (also called “goodness of fit test”)
8
1. Is there a relationship between the variables?
Visually (Stacked Bar Graph) Mathematically (statistical test) 2. If there is a relationship, how
STRONG is the relationship?
Two-way table contingency test (also called “test of independence”)
Format convention
Bad Presentation
Ex. Variables GENDER and VIEW OF LIFE Which sentence makes more sense? Does gender have an effect on View of life (Is life exciting , routine, or dull?) Does View of life (Is life exciting , routine, or dull?) have an effect on gender?
The independent variable is on the horizontal axis (X) The dependent variable is on the vertical axis (Y).
Independent variable: Gender Dependent variable: View of Life
We will study the differences in outcomes of the dependent variable (view of life) across the independent variable (gender). We will compare two groups (male and female) responses.
Good Presentation
No Relationship The 100%f stacked bar chart does
NOT significantly CHANGE for different categories of the IV.
Relationship The 100%f stacked bar chart DOES
significantly CHANGE for different categories of the IV. (at least one has to change for some relationship to be detected).
Dependent variable: Political Party
Independent variable: Area of residence
Is there a relationship/difference between the variables?
11
1. Is there a relationship between the variables?
Visually (Stacked Bar Graph) Mathematically (statistical test) 2. If there is a relationship, how
STRONG is the relationship?
Two-way table contingency test (also called “test of independence”)
Research Question: Is there a difference in house styles (DV) at different locations (IV)?
A sample is taken and organized into a
two way contingency table.
House Location
House Style Urban Rural Total
Split-Level 63 49 112
Ranch 15 33 48
Total 78 82 160
Is there a relationship between the variables?
Urban Rural
Split Split
100
80
60
40
20
Is there a significant difference?
Ranch Ranch
1. Form hypothesis.
2. Calculate the chi “kai” square (2) statistic.
3. Find the (2) significant value in the table.
4. Compare 2 statistic to 2 significant.
How to we test this claim?
total totalRow ColumnExpected count=
total number of cellscell
2
2
each cell
statistic
observed Expected
Expected
Ho “null hypothesis”: No relationship exists (variables are Independent) Ha “alternative hypothesis”: Relationship exists (variables are Dependent)
2significant = 2
α,df
=.05 (default value used in any statistical program) we will discuss this in detail soon!
df = (#rows-1)(# columns–1)
2Statistic > 2
Significant REJECT Ho. Accept Ha.
2Statistic < 2
Significant FAIL TO REJECT Ho.
Remember to state your result in the context of the specific problem!
Find the (2) significant value in the table
df = (#rows-1)
x (# columns–1)
Step 1: Form Hypothesis
Solution Step 2: Calculate chi-squared (2) statistic
Step 3: Find 2 significant
Step 4: Compare 2 statistic to 2 significant
Ho “null hypothesis”: No relationship exists (Variables are Independent)
Ha “alternative hypothesis”: Variables are related (Dependent)
df = (#rows-1)(# columns–1)
= (2 - 1)(2 - 1) = 1 =.05 (default value)
2significant= 2
α,df = 2.05,1 = 3.841
2 Statistic (7.62) > 2 Significant (3.841 )
Reject Ho. There is evidence of relationship between the variables.
total totalRow ColumnExpected Count =
total number of cellsfor each cell
2
2
each cell
2 2
2 2
63 55 49 57
55 57
15 23 33 257.62
23 25
statistic
observed Expected
Expected
Observed Expected
Style Urban Rural Total Style Urban Rural Total
Split 63 49 112 Split 55 57 112
Ranch 15 33 48 Ranch 23 25 48
Total 78 82 160 Total 78 82 160
Location Location
= 112·78 = 55 160
The chi-square statistic compares the observed count in each table to the count that would be expected under the assumption of no association between the variables.
Remember to state your result in the context of the specific problem! The style of house differs depending on the location.
The market research group for Alber’s Brewery of Tuscon, AZ, wants to know whether
preferences of beer type (light, regular, dark) differ among gender (male, female).
If beer preference is independent of gender, one advertising campaign will be initiated.
However, if beer preference depends on the gender of the beer drinker, the firm will
tailor its promotions to different target markets.
Your turn!
At the .05 level of significance, is there a statistically significant difference between beer preference for males and females? What about at the .01 level of significance?
M
Stacked bar graph
F
100%
75%
50%
25%
0
25% Light 43%
Light
25% Dark 14% Dark
50% Regular
43% Regular
A survey was conducted and the following
data was collected:
Are the variables related?
Step 1: Form Hypothesis
Your turn! Solution Step 2: Calculate chi-squared (2) statistic
Step 3: Find 2 significant
Step 4: Compare 2 statistic to 2 significant
Ho “null hypothesis”: No relationship exists (Variables are Independent)
Ha “alternative hypothesis”: Variables are related (Dependent)
df = (#rows-1)(# columns–1)
= (3 - 1)(2 - 1) = 2 =.05 (default value)
22,.05 = 5.991
22,.01= 9.210
total totalRow ColumnExpected Count =
total number of cellsfor each cell
2
2
each cell
2 2
2 2
2 2
20 27 40 37
27 37
20 16 30 23
16 23
30 33 10 14
33 14
6.604
statistic
observed Expected
Expected
REJECT Ho at the .05 level of significance (2 Statistic (6.604) > 2 Significant (5.991)). There is a difference in beer preferences for men and woman. More females prefer light beer than men. Men prefer regular beer over light/dark and females prefer light/regular over dark beer.
FAIL TO REJECT Ho at the .01 level of significance (2 Statistic (6.604) < 2 Significant (9.210)). There is not enough evidence to reject Ho. Any differences in cell frequencies could be explained by chance.
What is α “Significance Level”? (also called a type 1 error).
The significance level is: how often you are wrong The most common α value is .05. This means there is a 5% chance that we are wrong in our findings from testing the claim. Conversely, there is a 95% chance that we are correct. What should α be? It is subjective. Other common values for α are .01 (99% confident in our results) and .10 (90% sure of our results).
Verdict: “GUILTY” REJECT Ho. There is enough evidence to
convict (guilty).
Important wording of conclusion “REJECT” vs “FAIL TO REJECT”
Let’s use the legal system as an example. The defendant is on trial for murder. A person is presumed innocent until proven guilty.
Ho = Innocent Ha = Guilty We assume Ho to be true. If there is enough evidence to prove Ha then we REJECT Ho and Ha is true.
Verdict: “NOT GUILTY” FAIL TO REJECT Ho. We do not say “Accept Ho”. There is
NOT enough evidence to convict but we are not proving innocence.
How much evidence we need is related to how confident we want to be in our results. α (the level of significance) is how often we are wrong (also called type 1 error). Small Claims Court for endangerment of a child: Less evidence needed to convict, α=.05 means there is a 5% chance you are wrong. Casey Anthony verdict is Reject Ho (“GUILTY”) Jury for 1st degree murder: More evidence needed to convict: α =.01 means there is a 1% chance you are wrong. Casey Anthony verdict is Fail to Reject Ho (“NOT GUILTY”)
“Statistically Significant” The value of α used depends
on how confident you want to be in your results. What is “statistically significant” to one person might not be to another.
Statisticians Has to have at least a 95% chance of being true to be considered worth telling people about (why α=.05 is default for any statistical program).
Manager If something has a 90% chance of being true (α =.1), it is probably better to act as if it were true rather than false!
The jury finds the defendant “GUILTY”… But we are WRONG and an innocent person goes to jail!!! This is Type I error. If 𝜶=5%, there is a 5%
chance that this error will occur. That is, we are wrong to
REJECT Ho.
Unfortunately, neither the legal system nor statistical testing are perfect. Remember that Ho = Innocent and Ha = Guilty.
Type I and Type II errors
Which is worse? How you feel about it depends on the level of 𝜶 that you choose. α & β have an Inverse Relationship. If we increase α, we decrease β, and vice versa!
The jury finds the defendant “NOT GUILTY”… But we are WRONG and a guilty person is set free (ex.
OJ Simpson, Casey Anthony)!!! This is Type II error. If β=5%, there is a 5% chance
that this error will occur. That is, we are wrong to FAIL
TO REJECT Ho.
Maybe this will help you remember Type I and Type II errors
We will learn how to calculate Type II error in another lecture…
The chi-square distribution is defined by the degrees of freedom (df).
df = (# outcomes of row – 1) (# outcomes of column – 1)
Chi-Square (2) Distribution
As the number of possible outcomes of a variable increases, the curve approaches a normal distribution.
REJECT Ho FAIL TO REJECT Ho
Compare 2 statistic to 2 significant
or p-value to 𝜶
𝝌𝟐statistic > 𝝌𝟐significant
p-value < 𝜶 Reject Ho
p-value > 𝜶 𝝌𝟐statistic < 𝝌𝟐significant
Fail to Reject Ho
𝝌𝟐statistic < 𝝌𝟐significant p-value > 𝜶
Fail to Reject Ho
CHISQ.TEST Excel function
p value
p-value Statistical significance
>.10 Not Sig.
<.10 Marginal
<.05 Fair
<.01 Good
<.001 Excellent
Since .04 (p-value) < .05 (α) Reject Ho
The sample size is large (expected frequency of each cell is > 5)
Chi-square Assumptions
Your turn!
Yes! We satisfy the assumption. If we did not, we cannot trust the results of this test!
What if Assumptions are not met?
Possibly Combine categories to increase values in each cell!
Here, there are substantially fewer older adults than any other group. We could combine the middle age and older adult categories into a “not young” category. Then we would have 2x3 cross tab with larger n values.
Young Not Young Music 14 12 News 4 23 Sports 7 12
6
6
Fisher’s exact test can be used if E(x) <5…but only for 2x2 tables
28
1. Is there a relationship between the variables?
Visually (Stacked Bar Graph) Mathematically (statistical test) 2. If there is a relationship, how
STRONG is the relationship?
Two-way table contingency test (also called “test of independence”)
There are two families of effect sizes (r and d)
“d” family Quantifying the size of the difference between two groups
“r” family Measuring the association (CORRELATION) between the variables. How much can the change (variance) in one variable be explained by the other?
Effect size is a measure of the strength of a relationship
At the .05 significance level, Reject Ho. There is evidence of relationship between the variables implies that men and women have different beer preferences. So…
1. Which cell(s) caused the difference (difference between proportions)?
More females prefer light beer than men. Men prefer regular beer over light/dark and females prefer light/regular over dark beer.
2. Is the difference indicative of a weak, moderate, or strong relationship? (effect size)
Urban Rural
Split Split
100
80
60
40
20
How BIG is the difference?
Ranch Ranch
1. We are only concerned with effect size if the result of the (chi-square) test was statistically significant. 2. The size of the p-value is no indication of the strength of the association (ex. small P-value does not imply strong association) 3. We are covering only the most widely used statistical tools (there are still many more but this is a basic course in statistics and those tools are for another course).
Some important points…
Some common measures in the “r” family
Phi “Fi” (2x2 tables)
Cramer’s V (not 2x2 tables)
.40-.59 relatively strong
.60-.79 strong
General Guidelines for interpreting strength: A value of .1 is considered a weak (small) effect, .3 a moderate effect and .5 a strong (large) effect.
Phi 𝜙 2 7.62
0.22160n
For effect size strength, .20(sqrt(2-1))=.20, which is a small effect size. The relationship is WEAK!
The common measures in the “r” family
2 6.6040.15
150 2V
n df
Cramer’s V
Another WEAK relationship!
According to Cohen’s guidelines in the SPSS book (A Guide to Doing Statistics in Second Language Research Using SPSS, Table 4.8, p. 119) for effect size strength, w=.17(sqrt(3-1))=.24, which is a small to medium effect size.
Squaring phi will give you the variance that can be explained. Whether the house location is urban or rural explains (.22x.22=.05) 5% of the variance in the style of house built.
The percentage variance effect sizes
The gender of a person explains 20% of the variance in marital status.
For effect size strength, .20(sqrt(2-1))=.20, which is a small effect size. The relationship is WEAK!
The relationship is WEAK!
The relationship is WEAK!
Squaring Cramer’s V will give you the variance that can be explained. The gender of a person explains (.15x.15=.023) 2.3% of the variance in beer preference.
2
2
each cell
2 2
2 2
63 55 49 57
55 57
15 23 33 257.62
23 25
statistic
observed Expected
Expected
2
2
each cell
2 2
2 2
2 2
20 27 40 37
27 37
20 16 30 23
16 23
30 33 10 14
33 14
6.604
statistic
observed Expected
Expected
Evaluate difference for each category republican
rural vs suburban rural vs urban suburban vs urban
Independent rural vs suburban rural vs urban suburban vs urban
Democrat rural vs suburban rural vs urban suburban vs urban
Other rural vs suburban rural vs urban suburban vs urban
The chi-squared test shows a relationship exists…but where does the relationship (difference) occur?
(at least one has to change for some relationship to be detected). find where any significant associations may be in the r x c table by calculating adjusted residuals, and by partitioning the table according to the Lancaster-Irwin method.
• Standardized residual method:
• The chi-square test is an overall test to see if there are differences between any of the cell frequencies. If just two of the cells in the design are significantly different, the test will be significant. Only the Republican party showed significant change.
• If you find a SIGNIFICANT
relationship, you need to do a
post-hoc test to discover which
pairs of cells are significantly
different.
The “d” family (amount of difference)
2x2 table Not 2x2 table
Measure of effect size:
• Odds ratio (OR)
Measure of effect size:
• Adjusted standardized residuals
Urban Rural
Split Split
100%
80%
60%
40%
20%
How BIG is the difference?
M F
100%
75%
50%
25%
0
25% Light 43%
Light
25% Dark 14% Dark
50% Regular
43% Regular
The chi-squared test shows a relationship exists…but where does
the relationship (difference) occur?
Rural Rural
22 / 852.41
9 / 84no fluRR
Odds Ratio
63 33 20792.83
15 49 735
adOR
bc
“Group 1 had odds of having outcome 1 OR times (“more” if OR>1; “less” OR<1) than those who were in group 2”.
Urban locations had odds of having a split-level house style 2.83 times more than those who were in the rural area.
Vitamin C No Vitamin C Total
Cold 17 31 48
No cold 122 109 231
Total 139 140 279
Vaccine Placebo Total
Flu 9 22 31
No flu 75 63 138
Total 84 85 169
Treatment Thrombus No thrombus Total
Placebo 18 7 25
Aspirin 6 13 19
Total 24 20 44
Group 1 Group 2 Total
Outcome 1 a c a+c
Outcome 2 b d b+d
Total a+b c+d a+b+c+d
House Location
House Style Urban Rural Total
Split-Level 63 49 112
Ranch 15 33 48
Total 78 82 160
No universal agreement regarding what constitutes a ‘strong’ or ‘weak’ association: ◦ OR > 2.0 is ‘moderately strong’; OR > 5.0 is ‘strong’
Weak associations are more likely to be explained by undetected biases or confounders.
Hankinson SE et al. Obstet Gynecol. 1991;80:708-714.
Hildreth et al, 1981 Rosenberg et al, 1982 La Vecchia et al, 1984
Tzonou et al, 1984 Booth et al, 1989
Hartge et al, 1989 WHO, 1989
Wu et al, 1988 Prazzini et al, 1991
Newhouse et al, 1977 Casagrande et al, 1979
Cramer et al, 1982 Willet et al, 1981
Weiss, 1981 Risch et al, 1983
CASH, 1987 Harlow et al, 1988
Shu et al, 1989 Walnut Creek, 1981
Vessey et al, 1987 Beral et al, 1988
Odds Ratio 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Ho
sp
ital-
Based
C
ase-C
on
tro
l
Co
mm
un
ity-B
ased
C
ase-C
on
tro
l C
oh
ort
www.contraceptiononline.org
OR used to COMPARE STUDIES
Oral Contraceptive Use and Ovarian Cancer -ve Association + ve Association
Understanding the effect measures is important! You can compare studies (which you will do in the
Evidence-Based Practice course).
Consistency • Repeated observation of an association in studies
conducted on different populations under different circumstances
• If studies conducted by….different researchers; at different times; in different settings ; on different populations ; using different study designs;
……all produce consistent results, this strengthens the argument for causation
Out of every 10 people that smoke, on average 2 will likely get cancer.
People that smoke have odds of developing the cancer 3.9 times (390%) higher than those that don’t smoke
One would need to harm 5 patients ( with smoke) see one case of cancer develop.
Vitamin C No Vitamin C Total
Cold 17 31 48
No cold 122 109 231
Total 139 140 279
Question of interest: Is smoking related to cancer?
Vaccine Placebo Total
Flu 9 22 31
No flu 75 63 138
Total 84 85 169
Treatment Thrombus No thrombus Total
Placebo 18 7 25
Aspirin 6 13 19
Total 24 20 44
Those that smoke are 3 times (or 300%) more likely to develop lung cancer than those that don’t smoke.
/ ( ) 30 /1003
/ ( ) 10 /100
a a bRR
c c d
( / ( )) ( / ( ))
(30 /100) (10 /100) .2
ARR a a b c c d
There is a 200% increase in lung cancer cases for smokers compared to those that don’t smoke.
/ ( ) / ( ) (30 /100) (10 /100)2
/ ( ) 10 /100
a a b c c dRRR
c c d
1/ 1/ .2 5NNT ARR
30 903.9
70 10
adOR
bc
Measure of effect sizes for medical studies
Group 1 Group 2 Total
Outcome 1 a c a+c
Outcome 2 b d b+d
Total a+b c+d a+b+c+d
Odds ratio (OR)
Relative Risk
Relative risk improvement/reduction
Absolute risk improvement/reduction
Number needed to treat/harm (NNT/NNH)
ABSOLUTE vs RELATIVE Risk Remember our calculation for the smoking example:
Out of every 10 people that smoke, on average 2 will likely get cancer , 20% (ARI).
(X 3) or 300% more likely to develop lung cancer than those that don’t smoke (RR).
There is a 200% increase in lung cancer cases for smokers compared to those that don’t smoke (RRR). 30
100
10
100
Those that smoke and got cancer
Those that don’t
smoke and got cancer
Group A
Group B
RRI ARI
10% 30% 200% 20%
1% 3% 200% 2%
.1% .3% 200% .2%
There are different ways of describing the same risk which can profoundly affect how we perceive it. Ultimately, when deciding on whether to take a treatment, ideally you should decide with your doctor if the reduction in the ACTUAL (absolute risk) outweighs the risks, side-effects and costs of treatment.
The relationship between RR and RRR
The RRR sounds better for marketing purposes!...
Evaluate difference for each category republican
rural vs suburban rural vs urban suburban vs urban
Independent rural vs suburban rural vs urban suburban vs urban
Democrat rural vs suburban rural vs urban suburban vs urban
Other rural vs suburban rural vs urban suburban vs urban
The chi-squared test shows a relationship exists…but where does the relationship (difference) occur?
(at least one has to change for some relationship to be detected). find where any significant associations may be in the r x c table by calculating adjusted residuals, and by partitioning the table according to the Lancaster-Irwin method.
• Standardized residual method:
• The chi-square test is an overall test to see if there are differences between any of the cell frequencies. If just two of the cells in the design are significantly different, the test will be significant. Only the Republican party showed significant change.
• If you find a SIGNIFICANT
relationship, you need to do a
post-hoc test to discover which
pairs of cells are significantly
different.
The “d” family (amount of difference)
2x2 table Not 2x2 table
Measure of effect size:
• Odds ratio (OR)
Measure of effect size:
• Adjusted standardized residuals
Urban Rural
Split Split
100%
80%
60%
40%
20%
How BIG is the difference?
M F
100%
75%
50%
25%
0
25% Light 43%
Light
25% Dark 14% Dark
50% Regular
43% Regular
The chi-squared test shows a relationship exists…but where does
the relationship (difference) occur?
Rural Rural
Example: Income
Happiness Below Average Above Not much 208 131 49 Moderate 527 835 294 Very 185 454 272
Example: Income
Happiness Below Average Above Not much 208 (23%) 131 (9%) 49 (8%) Moderate 527 (57%) 835 (59%) 294 (48%) Very 185 (20%) 454 (32%) 272 (44%)
• Standardized residuals that have a positive value mean that the cell was over-represented in the actual sample, compared to the expected frequency, i.e. there were more subjects in this category than we expected.
• Standardized residuals that have a negative value mean that the cell was under-represented in the actual sample, compared to the expected frequency, i.e. there were fewer subjects in this category than we expected.
There is a significant difference across gender for WIDOWED only! The residuals can be deceiving,
you must use the standardized residuals.
A statistically significant relationship was found (chi-square test rejected Ho). SPECIFICALLY, there is a statistical difference in male and female responses for those that chose Light beer (look for values that are above +2 or below -2).
20 27
272.3
50 80(1 )(1 )(1 )(1 )
150 150row column
total total
O E
E
n n
n n
Add in statement…3.7 standard deviations…(way to phrase the 3.7 value).
Take out bottom part? Was confusing to explain
but maybe just becuase need statement.
To determine which of the categories are major contributors to the
statistical significance, the adjusted standardized residual is computed for each cell:
The “d” family for not 2x2 tables
2
2
each cell
2 2
2 2
2 2
20 27 40 37
27 37
20 16 30 23
16 23
30 33 10 14
33 14
6.604
statistic
observed Expected
Expected
male female
Light -2.3 3.5
Regular 0.9 -0.9
Dark 1.6 -1.9
adjusted
standardized
There are two contingency table tests:
(1) Two-way table contingency test (also called “test of independence”)
(2) One-way table contingency test (also called “goodness of fit test”)
42
1. Is there a relationship between the variable and a specific distribution?
Visually (Bar Graph) Mathematically (statistical test) 2. What is the effect size?
One-way contingency table test is called a “Goodness of Fit” test
A hypothesis is a belief about the results of a statistical study.
The “Goodness of Fit” tests if the outcomes of a variable
follows a hypothesized distribution (or put another way the
Relationship of variable to a specific distribution).
10
20
5
0
5
10
15
20
25
Democrat Republican Independent
12 12 11
0
5
10
15
20
25
Democrat Republican Independent
20
10
5
0
5
10
15
20
25
Democrat Republican Independent
“The Republican party is significantly larger than any other.”
“There is an equal amount of individuals in each party.”
“The Democrat party is significantly larger than any other.”
44
1. Is there a relationship between the variable and a specific distribution?
Visually (Bar Graph) Mathematically (statistical test) 2. What is the effect size?
Collected Raw
Data must be
organized into a
frequency table.
Seat f
Back 23
Middle 35
Front 29
Total 87
United Airlines compared these actual results to hypothesized results, which is the belief that fatality is the same whether one sits in the front, back, or middle of an airplane.
87/3 = 29
Front is 29 Middle is 29 Back is 29
This is a uniform distribution!
Example Is it safer to fly in the front, middle, or back of the airplane?
Matt McCormick, a survival expert for the National Transportation Safety Board, told Travel Magazine that “There is no one safe place to sit”.
In an effort to test this claim, United Airlines recorded the seat position for 87 fatalities.
1. Form hypothesis.
2. Calculate chi-squared (2) statistic.
3. Find 2 significant in table.
4. Compare 2 statistic to 2 significant.
How to we test this claim?
Ho “null hypothesis”: The data are consistent with a specified distribution.
Ha “alternative hypothesis”: The data are NOT consistent with a specified distribution.
2
2exp
exp
observed ected
ected
2,df
=.05 (default value) df = # outcomes– 1
2 statistic > 2 significant, Reject Ho, Accept Ha.
2 statistic < 2 significant, Fail to Reject Ho.
Step 1: Form Hypothesis
Solution Step 2: Calculate chi-squared (2) statistic
2
2
2 2 2
exp
exp
23 29 35 29 29 29
29 29 29
1.24 1.24 0 2.48
observed ected
ected
Step 3: Find 2 significant
Step 4: Compare 2 statistic
to 2 significant
Outcome f (observed)
Hypothesized distribution “f (expected)”
Back 23 29
Middle 35 29
Front 29 29
Total 87 87
Ho “null hypothesis”: The data are consistent with a specified distribution. There is no one
safe place to sit.
Ha “alternative hypothesis”: The data are NOT consistent with a specified distribution.
There is a safe place to sit.
df = # outcomes– 1 = 3 – 1 = 2 =.05 (default value) 2
,df = 2.05,2 = 5.991
2.48<5.991
2 statistic < 2 significant
Fail to Reject Ho.
There is not enough evidence to refute the claim that there is “no one safe place to sit!”.
Your turn! Goodness of Fit Is Sudden Infant Death Syndrome (SIDS) Seasonal? Data from King County, Washington regarding the number of deaths from SIDS for each season:
Season f
Winter 78
Spring 71
Summer 87
Fall 86
Total 322
Conclusion: Sudden infant death syndrome proportions across seasons are not statistically different from what’s expected by chance (i.e. all seasons being equal).
Step 1: Form Hypothesis
Step 2: Calculate chi-squared (2) statistic
Step 3: Find 2 significant
Step 4: Compare 2
statistic to 2 significant
df = # outcomes– 1 = 4 – 1 = 3 =.05 (default value) 2
,df = 2.05,3 = 7.815
2 statistic (2.10) < 2 significant (7.815)
Fail to Reject Ho.
2 2
2
2 2
78 80.5 71 80.5
80.5 80.5
87 80.5 86 80.5
80.5 80.5
2.10
statistic
Ho “null hypothesis”: Data follows hypothesized distribution (uniform – SIDS deaths for all seasons are the same. Ha “alternative hypothesis”: Data doesn’t follow the hypothesized distribution.
Season fo fe
Winter 78 322/4 =80.5
Spring 71 80.5
Summer 87 80.5
Fall 86 80.5
Total 322 322
CHISQ.TEST Excel function output compares p value to α
p value (.55)
𝝌𝟐statistic (2.10) < 𝝌𝟐significant (7.815) p value (.55) > α (.05) Fail to Reject Ho
Goodness of Fit Test Assumptions
Test Assumptions
Exact Binomial test (we didn’t learn by hand)
2 outcomes Samples up to n=1000
Chi-square test Large sample: E(x)>5
51
1. Is there a relationship between the variable and a specific distribution?
Visually (Bar Graph) Mathematically (statistical test) 2. What is the effect size?
.80-1.00 very strong departure from “fit”
.40-.59 relatively strong
.60-.79 strong
.40-.59 Relatively
strong departure from “fit”
0.00-.09 A perfect fit
.10-.19 small departure from “fit”
The interpretation is different from the test of independence!
.60-.79 Strong
departure from “fit”
.20-.39 Moderate departure from “fit”
Goodness of fit effect size Just like the test of Independence, use Cramer’s V
(more than 2 outcomes) or Phi (2 outcomes). Unlike the test of independence, compute the effect size whether you Reject or Fail to Reject Ho.
A value of .1 is considered a close to perfect fit
.3 a moderate effect
.5 a weak fit.
Goodness of fit effect size
Interpretation: A value of 0 indicates that the sample proportions are exactly equal (a perfect fit) to the hypothesized proportions (i.e., O = E). As v increases, the degree of departure from “a perfect fit” increases. Since V=.05, there is a small effect, or small departure from “fit”
2 2.10.05
322 3V
n df
Cramer’s V
Is Sudden Infant Death Syndrome (SIDS) Seasonal? Data from King County, Washington regarding the number of deaths from SIDS for each season:
Season f
Winter 78
Spring 71
Summer 87
Fall 86
Total 322
2 2
2
2 2
78 80.5 71 80.5
80.5 80.5
87 80.5 86 80.5
80.5 80.5
2.10
statistic
To test if a sample of data came from a population
with a specific distribution is called a GOODNESS-
OF-FIT TEST.
Why are we learning this?
We can use the GOODNESS-OF-FIT TEST to validate the use of a specific distribution for:
SIMULATION
PROBABILITY
Typical steps for a Statistical study
55
Reminder…How this lecture fits in with everything we have learned so far...
1. Define the goals • Research Question
2. Collect the data • Research Designs
3. Organize the data • Tables for each variable (f, %f, cf, %cf )
• Table for 2 qualitative variables (f contingency table, %f
total, %f independent (column) variable)
4. Present the data • Graphs of each variable (pareto pie, bar, ogive, histogram,
boxplot)
• Graphs for 2 qualitative variables (Stacked & Clustered bar
graph)
• Graph for 1 quantitative variable and 1 qualitative variable
(Comparative Boxplot )
5. Describe the data • Statistics and Parameters
6. Analyze the data • Statistical tests (chi-square, Fisher’s Exact)
• Effect size (OR, Adjusted Standardized Residual, Phi, Cramer’s V)
7. Interpret results
Remember If you need help…call me, see me, or email me.
End of the Lecture!