Transcript Inferential Statistics

SAA 2023
COMPUTATIONALTECHNIQUE FOR
BIOSTATISTICS
Inferential Statistics
Inferential Statistics
Computational Statistics:
- All you need to do is choose
statistic
- Computer does all other
steps for you!!!!!!
Inferential Statistics
Inferential Statistics
Inferential Statistics
Inferential Statistics
Inferential Statistics
For instance, if I am asked by a layperson what is meant
exactly by statistics, I will refer to the following Old
Persian saying: “Mosht Nemouneyeh Kharvar Ast
(translated: a handful represents the heap).
Inferential Statistics
This brief statement will describe, in one sentence, the
general concept of inferential statistics. In other words,
learning about a population by studying a randomly
chosen sample from that particular population can be
explained by using an analogy similar to this one.
Inferential Statistics
Involves using obtained sample
statistics to estimate the corresponding
population parameters
Most common inference is using a
sample mean to estimate a population
mean (surveys, opinion polls)
Drawing conclusions from sample to
population
Sample should be representative
Inferential Statistics
Inferential statistics allow us to make
determinations about whether groups
are significantly different from each
other.
Inferential Statistics
Inferential statistics is not a system of
magic and trick mirrors. Inferential
statistics are based on the concepts of
probability (what is likely to occur) and
the idea that data distribute normally
Inferential Statistics
Sample of
observations
Entire population of
observations
Random selection
Statistic
Parameter
µ=?
X
Statistical inference
Inferential Statistics
Inferential statistics is based on a strange
and mystical concept called falsification.
Although you might think the process is
simple
Write a hypothesis, test it, hope to prove it
Inferential statistics works this way:
Inferential Statistics
Write a hypothesis you believe to be true.
Write the OPPOSITE of this hypothesis,
which is called the null hypothesis.
Test the null, hoping to reject it.
- If the null is rejected, you have evidence
that the hypothesis you believe to be
true may be true.
- If the null is failed to reject, reach no
conclusion.
Inferential Statistics
Two major sources of error in research
Inferential statistics are used to make generalizations
from a sample to a population. There are two sources of
error (described in the Sampling module) that may result
in a sample's being different from (not representative of)
the population from which it is drawn.
These are:
Sampling error - chance, random error
Sample bias - constant error, due to
inadequate design
Inferential Statistics
Inferential statistics take into account sampling
error. These statistics do not correct for
sample bias. That is a research design issue.
Inferential statistics only address random
error (chance).
Sampling error - chance, random error
Sample bias - constant error, due to
inadequate design
Inferential Statistics
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Inferential Statistics
p-value
The reason for calculating an inferential statistic is
to get a p-value (p = probability). The p value is
the probability that the samples are from the
same population with regard to the dependent
variable (outcome).
Usually, the hypothesis we are testing is that the
samples (groups) differ on the outcome. The pvalue is directly related to the null hypothesis.
Inferential Statistics
The p-value determines whether or not we
reject the null hypothesis. We use it to
estimate whether or not we think the null
hypothesis is true.
The p-value provides an estimate of how
often we would get the obtained result by
chance, if in fact the null hypothesis were
true.
Inferential Statistics
If the p-value (< α- value) is small, reject
the null hypothesis and accept that the
samples are truly different with regard to
the outcome.
If the p-value (> α- value) is large, fail to
reject the null hypothesis and conclude
that the treatment or the predictor variable
had no effect on the outcome.
Inferential Statistics
Steps for testing hypotheses





Calculate descriptive statistics
Calculate an inferential statistic
Find its probability (p-value)
Based on p-value, accept or reject the null hypothesis (H0)
Draw conclusion
Inferential Statistics
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Inferential Statistics
One-Sample T: GPAs
Variable N Mean StDev SE Mean
90% CI
GPAs
50 2,9580 0,4101 0,0580 (2,8608; 3,0552)
Variable N Mean StDev SE Mean
95% CI
GPAs
50 2,9580 0,4101 0,0580 (2,8414; 3,0746)
Sample mean = 2,96
- 90% confident that μ (population mean) is between 2,86 and 3,06
- 95% confident that μ (population mean) is between 2,84 and 3,07
95% CI width > 90% CI width
The larger confidence coefficient, the greater the CI width
Inferential Statistics
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One-Sample T: GPAs
Test of μ = 3 vs < 3 (directional, one-tail to the right)
95% Upper
Variable N Mean StDev SE Mean
Bound
T
P
GPAs
50 2,9580 0,4101 0,0580
3,0552 -0,72 0,236
Test of μ = 3 vs ≠ 3 (non-directional, two-tail)
Variable N Mean StDev SE Mean
95% CI
T
P
GPAs
50 2,9580 0,4101 0,0580 (2,8414; 3,0746) -0,72 0,472
Test of μ = 3 vs > 3 (directional, one-tail to the left)
95% Lower
Variable N Mean StDev SE Mean
Bound
T
P
GPAs
50 2,9580 0,4101 0,0580
2,8608 -0,72 0,764
Inferential Statistics
One-Sample T: GPAs
Test of μ = 3 vs < 3
95% Upper
Variable N Mean StDev SE Mean
Bound
T
P
GPAs
50 2,9580 0,4101 0,0580
3,0552 -0,72 0,236
The test statistic, t = -0,72, is the number of std deviations that the sample mean
2,96 is from the hypothesized mean, μ = 3.
p-value is the probability that a random sample mean is less than or equal to
2,96 when Ho: μ = 3 is true.
The rejection region consists of all p-values less than α.
p-value = 0,236 > α = 0,05
Then, we fail to reject Ho
Conclusion: There is not sufficient evidence in the sample to conclude that the
true mean GPA, μ, is less than 3.00.
Inferential Statistics
Two sample Ttest
One-sample t-test
One-sample t-test
Inferential Statistics
Non-directional (2-tailed test): In this form of the test, departure can
be observed from either end of the distribution. Thus, no direction
for expected results are specified. The null and alternative
hypotheses are as follows:
Directional (1-tailed test): In this form of the test, the rejection region
lies at only one end of the distribution. The direction is specified
before any analysis begins.
One-sample t-test
Cholesterol Example: Suppose population mean μ is 211
x = 220 mg/ml
s = 38.6 mg/ml
n = 25 (town)
H0 : m = 211 mg/ml
HA : m ¹ 211 mg/ml
t0.05,24  2.064
X  0 220  211
t

 1.17
s / n 38.6 / 25
For an a = 0.05 test we use the critical value determined
from the t(24) distribution.
Since |t| = 1.17 < 2.064 (table t) at the a = 0.05 level
We fail to reject H0.
The difference is not statistically significant
Hypothesis Testing
One-sample t-test
 If the observation is
outside the standard, we
reject the hypothesis that
the sample is
representative of the
population
6
4
2
10
/9
0
20
/8
0
30
/7
0
40
/6
0
50
/5
0
60
/4
0
70
/3
0
80
/2
0
90
/1
0
99
/0
1
0
10
8
6
4
2
0
10
/9
0
20
/8
0
30
/7
0
40
/6
0
50
/5
0
60
/4
0
70
/3
0
80
/2
0
90
/1
0
99
/0
1
 We observe a sample and
infer information about
the population
8
10
8
6
4
2
0
10
/9
0
20
/8
0
30
/7
0
40
/6
0
50
/5
0
60
/4
0
70
/3
0
80
/2
0
90
/1
0
99
/0
1
 We set a standard beyond
which results would be
rare (outside the
expected sampling
error)
10
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Degree of Freedom (df)
Two-sample t-test
Two-sample t-test
Inferential Statistics
Non-directional (2-tailed test): In this form of the test, departure can
be observed from either end of the distribution. Thus, no direction
for expected results are specified. The null and alternative
hypotheses are as follows:
Directional (1-tailed test): In this form of the test, the rejection region
lies at only one end of the distribution. The direction is specified
before any analysis begins.
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
Two-sample t-test
N
Healthy 9
Cystic Fibrosis 13
Mean Std. Dev.
18.9
11.9
5.9
6.3
We test the hypothesis of equal means for the two
populations, assuming a common variance.
H0 : 1 = 2, HA : 1  2
( x1  x2 )  ( 1   2 )
t
~ t (df  n1  n2  2)
s x1  x2
Two-sample t-test
s
2
n

1
 1s12  n2  1s22
n1  n2  2 
9  1 5.9


 13  1  6.32
9  13  2 

sx1  x2
t
2

754.76   37.73
20 
1 1
1 1
 s (  )  37.7 * (  )  2.66
n1 n2
9 13
2
x1  x 2
sx1 x2

18.9  11.9

 2.63
2.66
t0.0520  2.086
| t | t0.0520 , reject H 0 , accept H A .
Two-sample t-test
MINITAB PROCEDURE – Comparing Two Population Means (RealEstate)
Stat ► Basic Statistics ► 2-Sample t
Choose Samples in one column
Samples: Select …………(SalePrice)
Subscripts: Select ………..(Location)
Click Graphs Check Boxplots of data OK
Click Options
Confidence level: Enter 90 OK OK
Boxplot of SalePrice
1000000
900000
Ho : μ1 = μ2 or μ1 - μ2 = 0
800000
HA : μ1 ≠ μ2 or μ1 - μ2 ≠ 0
SalePrice
700000
600000
500000
400000
300000
200000
100000
East
West
Location
Two-sample t-test
Result: Two-sample T for SalePrice
Location N Mean StDev SE Mean
East
34 312702 188163 32270
West
26 339143 255571 50122
Difference = μ (East) - μ (West)
Estimate for difference: -26442
90% CI for difference: (-126602; 73719)
T-Test of difference = 0 (vs not =): T-Value = -0,44 p-Value = 0,660 DF = 44
The difference in mean sale prices = $26,441
90% Confidence that the true mean difference in sale prices is in the interval
(-$126,602 < μ (East) - μ (West) < $73,719)
p-Value = 0,660 > α = 0,05/0,10
Fail to reject Ho
We do not have sufficient evidence in the sample to conclude that there are
significant differences in mean sale prices in markets east and west.
Paired t-test
 Test to deal with two observations with strong
comparability (e.g. two treatments on the
same individuals, or one individual Before vs.
After treatment, very close plots)
 Sample 1. X11, X12, …, X1n
 Sample 2. X21, X22, …, X2n
 Method:
Calculate differences between two measurements
for each individual di = Xi1 – Xi2
Calculate
2
d

d 
n
j
, sd 
 (d
j
d)
n 1
, sd 
sd
n
Paired t-test
H 0 :  d  0, H A  0.
d  d
d
t

~ t ( n  1)
sd
sd
n
Paired t-test
Test infection of virus on tobacco leaves. Number of
death pots on leaves.
Virus 1
X1j
Plant
1
2
3
4
5
6
7
8
Total
Mean
9
17
31
18
7
8
20
10
120
15
H 0 : d  0, H A  0.
Virus 2
X2j
dj
10
11
18
14
6
7
17
5
88
11
-1
6
13
4
1
1
3
5
32
4
  0.05, t0.05( 7 )  2.365
d 4
(1) 2  6 2  ...  52  (32) 2 / 8
sd 
8 1
 4.31
4.31
sd 
 1.52
8
4
t
 2.632  t0.05
1.52
Paired t-test
Advantages:
2
2
1). Usually sd  sx1  x2 , it is easy to find
a true small difference.
2). Do not need to consider if the
variances of two populations are same
or not.
Paired t-test
Hypothesis Testing
Hypothesis testing is always a fivestep procedure:
Formulation of the null and the
alternative hypotheses
Specification of the level of significance
Calculation of the test statistic
Definition of the region of rejection
Selection of the appropriate hypothesis
Hypothesis Testing
3 Steps to carry out a Hypothesis Test
1. Specify the 5 required elements of a
Hypothesis test listed above.
2. Using the sample data, compute either the
value of the test statistic or the p-value
associated with the calculated test
statistic.
Hypothesis Testing
3 Steps to carry out a Hypothesis Test
3.
Use one of three possible decision rules (don't mix
them up!):
1. Table method: Compare the calculated value with a
table of the critical values of the test statistic
 If the absolute (calculated) value of the test statistic 
to the critical value from the table, reject the null
hypothesis (HO) and accept the alternative hypothesis
(HA).
 If the calculated value of the test statistic < the critical
value from the table, fail to reject the null hypothesis
(H0).
Hypothesis Testing
Hypothesis Testing
Hypothesis Testing
3 Steps to carry out a Hypothesis Test
3. Use one of three possible decision rules
(don't mix them up!):
2. Graph method: Compare the calculated value
with the t-distribution graph of the test statistic
 Reject the NULL hypothesis if the test statistic
falls in the critical region. Fail to reject the
NULL if the test statistic does not fall in the
critical region.
Hypothesis Testing
Rejection region at the α=10% significance level.
Hypothesis Testing
Rejection region at the α=5% significance level.
Hypothesis Testing
Rejection region at the α=10% significance level.
Hypothesis Testing
3 Steps to carry out a Hypothesis Test
3. Use one of three possible decision rules
(don't mix them up!):
3. p-value method: Reject the NULL hypothesis
if the p-value is less that α. Fail to reject the
NULL if the p-value is greater than α.
 The exact p-value can be computed, and if p <
0.05, then H0 is rejected and the results are
declared statistically significant . Otherwise, if p
 0.05 then H0 is failed to reject and the results
are declared not statistically significant .
Hypothesis Testing
Hypothesis Testing
Example of hypothesis testing can be found
in the jury system. There are three party
involved in a court case. i.e. plaintiff
(prosecutors), defendant (accuse) and the
judges.
The judge will form a hypothesis as below
before hearing a case;
Hypothesis
Ho : The evidences are not significantly
strong enough to proof the defendant guilty
H1 : The evidences are significantly strong to
proof defendant guilty
Hypothesis Testing
The main function of the plaintiff play in a
case is to continuously supply strong
evidence to proof the defendant guilty.
Whereas the function of the defendant is to
defend himself by rejecting the evident
provide by the plaintiff.
The judges role is to collect information
supplied by the plaintiff and defendant and
make decision about the hypothesis validity.
Hypothesis Testing
To perform a hypothesis test, we start with two
mutually exclusive hypotheses. Here’s an
example: when someone is accused of a crime,
we put them on trial to determine their innocence
or guilt. In this classic case, the two possibilities
are the defendant is not guilty (innocent of the
crime) or the defendant is guilty. This is
classically written as:
H0: Defendant is Innocent ← Null Hypothesis
HA: Defendant is Guilty ← Alternate Hypothesis
Hypothesis Testing
Hypothesis Testing
Unfortunately, our justice systems are not
perfect. At times, we let the guilty go free and
put the innocent in jail. The conclusion drawn
can be different from the truth, and in these
cases we have made an error. The table below
has all four possibilities. Note that the
columns represent the “True State of Nature”
and reflect if the person is truly innocent or
guilty. The rows represent the conclusion
drawn by the judge or jury.
Type I and Type II Errors
Type I and Type II Errors
Alpha and Beta
Type I and Type II Errors
Type I and Type II Errors
NULL HYPOTHESIS
TRUE
FALSE
Reject the null Type I error
α
CORRECT
hypothesis
Rejecting a true
null hypothesis
Fail to reject the
CORRECT
null hypothesis
Type II error
β
Failing to reject a
false null hypothesis
Type I and Type II Errors
Two of the four possible outcomes are
correct. If the truth is they are innocent and
the conclusion drawn is innocent, then no
error has been made. If the truth is they are
guilty and we conclude they are guilty,
again no error. However, the other two
possibilities result in an error.
Type I and Type II Errors
A Type I (read “Type one”) error is when
the person is truly innocent but the jury
finds them guilty. A Type II (read “Type
two”) error is when a person is truly guilty
but the jury finds him/her innocent. Many
people find the distinction between the
types of errors as unnecessary at first;
perhaps we should just label them both as
errors and get on with it.
Type I and Type II Errors
However, the distinction between the two
types is extremely important. When we
commit a Type I error, we put an innocent
person in jail. When we commit a Type II
error we let a guilty person go free. Which
error is worse?
Type I and Type II Errors
The generally accepted position of society
is that a Type I Error or putting an innocent
person in jail is far worse than a Type II
error or letting a guilty person go free. In
fact, the burden of proof in criminal cases
is established as “Beyond reasonable
doubt”.
Another way to look at Type I vs. Type II
errors is that a Type I error is the
probability of overreacting and a Type II
error is the probability of under reacting.
Type I and Type II Errors
In statistics, we want to quantify the
probability of a Type I and Type II error.
The probability of a Type I Error is α (Greek
letter “alpha”) and the probability of a Type
II error is β (Greek letter “beta”). Without
slipping too far into the world of theoretical
statistics and Greek letters, let’s simplify
this a bit. What if the probability of
committing a Type I error was 20%?
Type I and Type II Errors
A more common way to express this would
be that we stand a 20% chance of putting
an innocent man in jail. Would this meet
your requirement for “beyond reasonable
doubt”? At 20% we stand a 1 in 5 chance
of committing an error. This is not sufficient
evidence and so cannot conclude that
he/she is guilty.
Type I and Type II Errors
The formal calculation of the probability of
Type I error is critical in the field of
probability and statistics. However, the
term "Probability of Type I Error" is not
reader-friendly. For this reason, the phrase
"Chances of Getting it Wrong" is used
instead of "Probability of Type I Error".
Type I and Type II Errors
Most people would agree that putting an
innocent person in jail is "Getting it Wrong"
as well as being easier for us to relate to.
To help you get a better understanding of
what this means, the table below shows
some possible values for getting it wrong.
Type I and Type II Errors
Chances of Getting it Wrong
(Probability of Type I Error)
Percentage
Chances of sending an innocent man to jail
20% Chance
1 in 5
5% Chance
1 in 20
1% Chance
1 in 100
.01% Chance
1 in 10,000
Controlling Type I and Type II Errors
 , , and n are interrelated. If one is kept constant,
then an increase in one of the remaining two will
cause a decrease in the other.
 For any fixed , an increase in the sample size n will
cause a ??????? in 
 For any fixed sample size n , a decrease in  will
cause a ??????? in .
 Conversely, an increase in  will cause a ???????
in  .
 To decrease both  and , ??????? the sample
size n.
Planning a study
Suppose
you
were
interested
in
determining whether treatment X has an
effect on outcome Y—there are several
issues that need to be addressed so that a
sound inference can be made from the
study result
Planning a study
What is the population?
How will you select a sample that is
representative of that population?
There are many ways to produce a sample, but
not all of them will lead to sound inference
Sampling strategies
Probability samples—result when subjects
have a known probability of entering the
sample
Simple random sampling
Stratified sampling
Cluster sampling
Sampling strategies
Probability samples can be made to be
representative of a population
Non-probability samples may or may not
be representative of a population—it may
be difficult to convince someone that the
sample results apply to any larger
population
Planning a study
Clinical trials are generally designed to be
efficacy trials—highly controlled situations
that maximize internal validity
We want to design a study to test the
effect of treatment X on outcome Y, and try
to make sure that any difference in Y is
due to X
Planning a study
 At the end of this study you observe a difference
in outcome Y between the experimental group
and the control group.
 All of the effort in designing the study with strict
control is for one reason—at the end of the study
you want only two plausible explanations for the
observed outcome
Chance
Real effect of treatment X
Planning a study
 The reason you want only these two
explanations is because if you can rule out
chance, you can conclude that treatment X must
have been the reason for the difference in
outcome Y
 All inferential statistical tests are used to
estimate the probability of the observed outcome
assuming chance alone is the reason for the
difference.
 If there are multiple competing explanations for
the observed result, then ruling out chance
offers little information about the effectiveness of
treatment X
Inferential statistics
Hypothesis testing—answering the
question of whether or not treatment X
may have no effect on outcome Y
Point estimation—determining what the
likely effect of treatment X is on outcome Y
Hypothesis testing
The goal of hypothesis testing is
somewhat twisted — it is to disprove
something you don’t believe
In this case you are trying to disprove that
treatment X has no effect on outcome Y
You start out with two hypotheses
Hypothesis testing
Null Hypothesis (HO)
Treatment X has no effect on outcome Y
Alternative Hypothesis (HA)
Treatment X has an effect on outcome Y
Hypothesis testing
 If the trial has been carefully controlled, there
are only two explanations for a difference
between treatment groups—efficacy of X,
and chance
 Assuming that the null hypothesis is correct,
we can use a statistical test to calculate that
the observed difference would have occurred.
This is known as the significance level, or pvalue of the test.
Hypothesis testing
P-value
The probability of the observed outcome,
assuming that chance alone was involved in
creating the outcome.
In other words,
assuming the null hypothesis is correct, what is
the probability that we would have seen the
observed outcome.
This is only meaningful if chance is the only
competing plausible explanation.
Hypothesis testing
If the p-value is small, meaning the
observed outcome would have been
unlikely, we will reject that chance played
the only role in the observed difference
between groups and conclude that
treatment X does in fact have an effect on
outcome Y
How small is small?
Hypothesis testing
Reality ->
Decision
Retain HO
Reject HO
HO is true
HO is false
Correct
Decision
Type II Error
()
(.2, .1)
Correct
Decision
Type I Error
()
(.05, .01)
Hypothesis testing
 Power analysis is used to try to minimize Type II
errors.
 Power (1-) is the probability of rejecting the null
hypothesis when the effect of X is some
specified value other than zero.
 Usually one specifies an expected effect and
uses power analysis to calculate the sample size
needed to keep  below some value (.2 is
common)
Point estimation
 Hypothesis testing can only tell you whether or
not the effect of X is zero, it does not tell you
how large or small the effect is.
 Important — a p-value is not an indication of the
size of an effect, it depends greatly on sample
size
 If you want an estimate of the actual effect, you
need confidence intervals
Point estimation
 Confidence intervals give you an idea of what
the actual effect is likely to be in the population
of interest
 The most common confidence interval is 95%
and gives an upper and lower bound on what
the effect is likely to be.
 The size of the interval depends on the sample
size, variability of the measure, and the degree
of confidence you want that the interval contains
the true effect.
Point estimation
 Many people prefer confidence intervals to
hypothesis testing, because confidence
intervals contain more information
 Not only can you tell whether the effect could
be zero (is zero contained in the interval of
possible effect values?) but you also have the
entire range of possible values the effect
could be
 So, a confidence interval gives you all the
information of a hypothesis test and a whole
lot more.
Choosing the right test
Typically one is interested in comparing
group means.
If the outcome is continuous, and one
independent variable:
Two groups — t-test
Three or more groups -- ANOVA
Choosing the right test
If the outcome is continuous and there is
more than one independent variable:
ANOVA, if all independent variables are
categorical
ANCOVA or multiple linear regression, if some
independent variables are continuous
Logic of Hypothesis testing
The further the observed value is from the
mean of the expected distribution, the
more significant the difference
Steps in test of Hypothesis
1. Determine the appropriate test
2. Establish the level of significance:α
3. Determine whether to use a one tail or
two tail test
4. Calculate the test statistic
5. Determine the degree of freedom
6. Compare computed test statistic against
a tabled value
3. Determine Whether to Use One
or Two Tailed Test
If the alternative hypothesis specifies
direction of the test, then one tailed
Otherwise, two tailed
Most cases
5. Determine Degrees of Freedom
Number of components that are free to
vary about a parameter
Df = Sample size – Number of parameters
estimated
Df is n-1 for one sample test of mean
Common Inferential Stats
T-Test -compare means of two groups
interval/ratio level of measurement
independent samples t-test
dependent or paired samples
Common Inferential Stats
ANOVA - analysis of variance􀂄
more than 2 means to compare or more
than 2 testing of means􀂄
interval/ratio level of measurement
Chi-square (x2) -testing hypothesis about
number of cases that fall into various
categories
nominal/ordinal level of measurement
Summary
 Descriptive stats summarize measures of central tendency and
variability􀂄
 Inferential determine how likely it is that results based on sample are
the same in population􀂄
 Must know level of measurement of variables to choose correct
 Parametric and non-parametric two types of statistics requiring
analysis of assumptions
 􀂄Pearson r, t tests and ANOVA examples of parametric
 􀂄Pearson r measure relationship or association between 2 variables
 T test determines if there is a significant difference between 2 group
means
 ANOVA determines if there is a significant difference between 3 or
more means
 X2 non-parametric statistic to assess relationship between 2
categorical variables
Two types of ANOVA
1.Independent groups - two different sets
of individuals In the graphic below, college
students are randomly assigned to Groups
1 and 2.
Two types of ANOVA
Example: Researchers are interested in exam anxiety. They administer
an anxiety inventory to students just before the final exam in a
Sociology class. They also adminster it before the final exam in a
Political Science class. To compare the two sets of scores, they use
either ANOVA or t-test for independent samples (hand
calculation).
Two types of ANOVA
2.Paired samples (sometimes referred to as Repeated
Measures or With Replication) - either the same
individuals or from matched groups (i.e., matched on
everything but the treatment (level of the Independent
variable).
Two types of ANOVA
Example: Researchers are interested in exam anxiety. They administer
an anxiety inventory on the second day of class. Then they give it
again on the day of the midterm. To compare the two sets of scores,
they use either ANOVA with replication or t-test for paired samples
(hand calculation).