Right statistics for the right research questions

Answering Research Question
with the Right Statistics

Types of categorical and quantifiable data
Data
Categorical Quantifiable
Nominal Ordinal Interval Ratio

What is categorical data?
• Data which cannot be quantified numerically
BUT
• Place into sets or categories (nominal data) or
ranked in some way (ordinal data)

What is quantifiable data
• Data can be measured numerically
• More precise
• Consist of interval data and ratio data

Four kinds of measurement scales
• Nominal
• Ordinal
• Interval
• Ratio

Nominal data
• A name value or category with no order or
ranking
Example:-
• Type of school
• Types of teaching method
• Gender
• Race

Ordinal data
• When categories are ordered, the variable is
known as an ordinal variable.
• Ordinal data tell us not only that things have
occurred, but also the order in which they
occurred.

Example
How of often you felt like insulting a student
(Please tick one)
• Every day
• Once a week
• Sometimes
• Never

Other examples of ordinal data
• Questions that rate the quality of students’
performance (for example, very good, good,
fair, poor)
• Agreements of attitude towards science
(Strongly agree, Agree, Disagree, Strongly
disagree)

Interval data
• Numerical values are assigned along an
interval scale with
• Equal intervals
• There is no zero point where the trait being
measured does not exist

Number of students scoring within
various ranges in IQ test
Scores Frequency
76-80 1
81-85 0
86-90 4
91-95 10
96-100 21
101-105 25
106-110 48
111-115 18
116-120 11

Other examples of interval data
• Temperature
• Blood pressure

Ratio data
• Same characteristics with interval data
BUT
• There is an absolute zero that represent some
meaning
Example:-
Costs, sales, number of students, number of
teachers,

Example of the scoring data
Students should be given an opportunity to
select a school of their choice
• Strongly agree _____
• Agree _____
• Disagree _____
• Strongly Disagree _____

A numeric score (or value) to each
response category
• Strongly agree 4
• Agree 3
• Disagree 2
• Strongly Disagree 1

Other example of scoring data
How of often you felt like insulting a student
(Please tick one)
• Every day
• Once a week
• Sometimes
• Never

response category
• Every day 4
• Once a week 3
• Sometimes 2
• Never 1

An example of multiple choice
question
• The quantity of charge which passes through
a circuit is measure in
A. Amps
B. Volts
C. Coulombs *
D. Watts

response category
• Correct response- 1 mark,
• Incorrect response- 0 mark
A. Amps 0
B. Volts 0
C. Coulombs 1
D. Watts 0

What is the level of measurement
of the following variables?
• The number of downloads of different bands’ songs on iTunes
• The names of the bands that were downloaded.
• The position in the iTunes download chart.
• The money earned by the bands from the downloads.
• The weight of bags brought by the passengers into the flight.
• The type of bags brought by the passengers into the flight.
• The phone numbers that the bands obtained because of their
fame.
• The gender of the people giving the bands their phone numbers.
• The instruments played by the band members.
• The time they had spent learning to play their instruments.

• Survey research (where we observe what
naturally goes on in the world without directly
interfering with it)
• Experimental research (where we manipulate
one variable to see its effect on another)

Two Main Types of Research
Variables
• Independent variableIndependent variable
• Dependent variableDependent variable

Independent variables
• An activity of characteristic believed to make a
difference with respect to some behavior
• Also known as experimental variable, cause,
treatment

Dependent variables
• The change or difference occurring as a result
of the independent variable
• Also known as criterion variable, effect,
outcome, posttest

Survey Research
• Survey research is the method of gathering
data from respondents thought to be
representative of some population, using
an instrument composed of closed
structure or open-ended items (questions).
• The goal is to describe some aspect or
characteristics (opinions, knowledge,
attitudes, beliefs) of a population.

Types of Surveys
• Cross-sectional
• Longitudinal
• Causal-comparative research

Cross-sectional
• Collects information from a sample of the
population at a specific point in time (“one
and done”)
• Information is collected at just one point in
time, even though it could take days to gather
all the data

Longitudinal
• Information is collected at different points
in time in order to study change over time
– Graduate students typically don’t do this
type of research because they want to
graduate

Causal-comparative
• To determine the cause for, or consequences
of, existing differencesexisting differences in groups of
individuals
• Also referred to as ‘ex post factoex post facto’ research
(Latin for ‘after the fact’) – retrospection
• Non-experimental research

Differences in causal-comparison
and experimental studies
• Causal-comparative studies
– the researcher cannot manipulate the
independent variable
• Experimental studies
– the researcher manipulates the independent
variable

“Experimental” Research
• Types
– Experimental
– Quasi-Experimental

Characteristics of
Experimental Research
• There is a control or comparison
group
• Subjects are randomly assigned to
groups

Characteristics of Quasi-
Experimental Research
• There is a control or comparison
group
• Intact groups are used

Diagramming
The Experimental/ Quasi Experimental
Research
• To illustrate research designs, a number
of symbols are used
– X1 = Treatment
– X2 = Control Group
– O = Observation (pretest or posttest)

Single-Group Pretest-Treatment-
Posttest Design
R O X1 O
This means subjects are randomly
assigned to a group, which is then
given a pretest, then there is a
treatment, then there is a posttest.
This means subjects are randomly
assigned to a group, which is then
given a pretest, then there is a
treatment, then there is a posttest.

Two-Group Pretest-Treatment-
Posttest Design.
• Have a control group and use
randomization.
R O X1 O
R O X2 O
R O X1 O
R O X2 O

Two-Group Treatment-Posttest-
Only Design
R X1 O
R X2 O
R X1 O
R X2 O

Solomon 4-Group Design
R O X1 O
R X1 O
R O O
R O
R O X1 O
R X1 O
R O O
R O

Quasi-Experimental Designs
X1 O
X2 O
X1 O
X2 O
The absence of R indicates there
is no random assignment.
Sometimes you will see a dotted
line between the two groups. This
indicates the two groups may not
be equivalent.

Quasi-Experimental Designs
• Pretest-Posttest Nonequivalent Group
Design
O X1 O
O X2 O
O X1 O
O X2 O

Time Series Designs
O O O X1 O O OO O O X1 O O O

What is the purpose of inferential
statistics?
• To compare two or more groups on the
independent variable in terms of the
dependent variable ( for example: “Is there
a significant difference between boys and
girls on self esteem?”)
Independent variable: gender (boys and
girls
Dependent variable: self esteem

Inferential statistics involves
hypothesis testing
• Null hypothesis: There is no significance
difference between boys and girls on self
esteem
• Alternative hypothesis: There is a significant
difference between boys and girls on self
esteem

Other purpose of inferential statistics
• Relate two or more variables (for example:
Does self esteem relate to academic
achievement?)
• Null hypothesis: There is no significant
relationship between self esteem and
academic achievement
• Alternative hypothesis: There is a
significant relationship between self
esteem and academic achievement

Important Perspectives
• Inferential statistics
– Allow researchers to generalize to a population
of individuals based on information obtained
from a sample of those individuals
– Assess whether the results obtained from a
sample are the same as those that would have
been calculated for the entire population

Types of Inferential Statistics
• Two issues discussed
– Steps involved in testing for significance
– Types of tests

Steps in Statistical Testing
• State the null and alternative hypotheses
• Set alpha level
• Identify and compute the test statistic
• Compare the computed test statistic to the
criteria for significance
Objectives 20.1 – 20.9

Alpha Level
• An established probability level which serves
as the criterion to determine whether to
accept or reject the null hypothesis
• Common levels in education
– .01
– .05 (the most common)
– .10

The null hypothesis
• If the probability values (Alpha level) is
less than or equal to the significance
level, then reject the null hypothesis
• If the probability values is greater than
the significance level, then fail to reject
the null hypothesis

Examples of Statistical Test
• T-test (independent samples)
• Analysis of variance
• Analysis of covariance
• Multiple analysis of variance
• Multiple analysis of covariance
• Chi-Square
• Pearson product moment correlation
• Multiple regression

Examples of Statistical Test
• Spearman rank-order correlation
• Point biserial correlation
• Phi coefficient

Independent samples t-test
• Types of hypothesis/ question: Group
comparison
• Number of independent variables: 1
• Number of dependent variables: 1
• Number of covariates: 0
• Type of data for independent variable:
Categorical
• Type of data for dependent variable:
Continuous

One way analysis of variance
(ANOVA)
comparison
• Number of independent variables: 1
Categorical
Continuous

Analysis of covariance (ANCOVA)
comparison
• Number of independent variables: 1 or more
Categorical
Continuous

Multiple analysis of variance
(MANOVA)
comparison
• Number of dependent variables: 2 or more
Categorical
Continous

Multiple analysis of covariance
(MANCOVA)
comparison
• Number of dependent variables: 2 or more
• Number of covariates: 1 or more
Categorical
Continuous

Pearson product moment correlation
• Types of hypothesis/ question: Relate
variables
• Number of variables: 2
• Type of data for variables: One variable is
continuous and another variable is continuous

Multiple regression
• Types of hypothesis/ question: Relate
variables in form of prediction
Continuous
Continuous

Let us apply in the research
setting

Ask your self
• Can you match/decide
• Research problem with
• Research objective with
• Research hypothesis with
• Research methodology with
• Numbers and characteristics of the sample
with
• Types of statistical analysis

• Aim of the research: Compare the attitude
towards learning between boy and girl
• Research question:
Is there any significant difference between
boys and girls on attitude towards learning?
• The null hypothesis:
There is no significant difference between
boys and girls on attitude towards learning
• The alternative hypothesis:
There is a significant difference between boys
and girls on attitude towards learning

• Independent variable: 1(Gender)
• Dependent variable: 1(Attitude towards
learning)
Categorical
Continuous
• Statistical test: ?

• Aim of the research: Compare the leadership
skills between principals from Smart School,
Boarding School and Cluster School
principals from Smart School, Boarding School
and Cluster School on leadership skills?

principal between principals from Smart
School, Boarding School and Cluster School on
leadership skills
There is a significant difference between
principals from Smart School, Boarding School
and Cluster School on leadership skills

• Independent variable: 1(Type of school where
the principal works)
• Dependent variable: 1 (Leadership skills)
Categorical
Continuous

• Aim of the research: Compare the level of
stress between male and female teachers
male and female teachers on stress?
male and female teachers on stress
There is a significant difference between male
and female teachers on stress

• Independent variable: 1(Gender)
• Dependent variable: 1(Stress)
Categorical
Continuous

• Aim of the research: To study whether
creative thinking skill or critical thinking skill is
the best predictor of academic achievement:
Which is the best predictor of academic
achievement: creative thinking skill or critical
thinking skill?

There is no significant contribution of all
predictor variables which are creative thinking
skill and critical thinking skill towards variation
in students’ academic achievement
There is a significant contribution of all
predictor variables which are creative thinking
skill and critical thinking skill towards variation
in students’ academic achievement

• Independent variable: 2 (Creative and critical
thinking skills)
• Dependent variable: 1(Academic
achievement)
Continuous
Continuous

• Aim of the research: Compare the academic
achievement between students who learn
maths using the computer courseware and
students who learn maths without using the
computer courseware
students who learn maths using computer
courseware and students who learn maths
without using the computer courseware on
mean scores of math post test after the effect
mean scores of math pre test is controlled?

mean scores of math pre test is controlled
There is a significant difference between
mean scores of math pre test is controlled

• Independent variable: 1 (Types of learning
method)
• Dependent variable: 1 (Mean scores of math
posttest)
• Covariate: Mean scores of math pretest
Categorical
Continuous
• Statistical test: Analysis of Covariance
(ANCOVA)

The interpretation of the
statistical outputs

Independent samples t-test
when you want to compare the
mean
scores of two different groups of
people or conditions

• Is the movie Scream 2 scarier than the original
Scream?
• How scary the movie is will be measured by
heart rates (which indicate anxiety) during
both films
• Use independent samples t-test to answer the
research question

• Does listening to music while you work
improve your work?
• Get some people to write an essay while
listening to their favourite music, and then
another group of people write an essay when
working in silence (this is a control group).
• Compare the essay marks by using
independent samples t-test

Hands-on exercise
• There are 12 spider-phobes who were
exposed to a picture of a spider and 12
different spider-phobes who were exposed to
a real-life tarantula (the groups are coded
using the variable group)
• Their anxiety was measured in each condition
(anxiety).

Research Question
• Is there any significant difference on the
anxiety between spider-phobes who were
exposed to a picture of a spider and spider-
phobes who were exposed to a real-life
tarantula?

• For these data, Levene’s test is non-significant
(because p = .386, which is greater than .05)
and so we should read the test statistics in the
row labelled Equal variances assumed.
• Had Levene’s test been significant, then we
would have read the test statistics from the
row labelled Equal variances not assumed.

Analysis of Variance
(ANOVA)
comparing the mean scores of more
than two groups

objective measure of libido between one
group of patients which receive a placebo,
one group of patients which receive a low
dose of Viagra and one group of patients
which receive a high dose of Viagra?

Analysis of Covariance
(ANCOVA)
extension of ANOVA that allows you
to explore differences between
groups while statistically controlling
for an additional (continuous)
variable

objective measure of libido between one
group of patients which receive a placebo,
one group of patients which receive a low
dose of Viagra and one group of patients
which receive a high dose of Viagra after the
effect of the partner’s libido is controlled?

Data file
• ViagraCovariate.sav

Two way ANOVA
Allows us to look at the individual
and joint effect of two independent
variables on one dependent variable

Two Way ANOVA
• Analysis of data which involve factorial design
• What is factorial design?

Factorial design
• When two or more independent variables
involved in a study

Example
Method A Method B
High ability
Low ability
2 X 2 Factorial Design

2 ways ANOVA
• Determine main effect on achievement for
method (determine there is a significant
difference between mean scores of Method A
and Method B)

2 ways ANOVA
• Determine main effect on achievement for
ability (determine there is a significant
difference between mean scores of high and
low ability)

Interaction effect
• Is there a significant interaction effect
between method and ability on achievement?

How to understand there is an
interaction effect between method
(method A and method B) and
students ability (high and low?

Hands-on Exercise
• Does gender moderate the relationship
between age-group and optimism?
OR
• Is there any significant interaction effect
between age-group and gender on the
optimism?

Multiple Analysis of Variance
(MANOVA)

• Multivariate analysis of variance (MANOVA) is
an extension of analysis of variance for use
when you have more than one dependent
variable.
• These dependent variables should be related
in some way, or there should be some
conceptual reason for considering them
together.

Example of research question
• Are males better adjusted than females in
terms of their positive and negative mood
states and levels of perceived stress?
OR
• Is the any significant difference on the linear
combination of positive mood, negative mood
and perceived stress between males and
females?

Data
• Data file survey3ED.sav

• When a subject is tested on the same variable
over time, it is a repeated measures design.

One-Way ANOVA with
Repeated Measures

• An investigator is interested in studying how
exposure to different levels of temperature
influences problem-solving ability.
• Thus, each subject was required to solve a set
of mathematical problems under the four
temperature conditions of 35°C, 30°C, 25°C,
and 20°C.

• The investigator expected that fewer errors
will be made when the room temperature is
dropping by 5°C every 5 min

Research Question
• Is there any significant differences on the
respondents’ problem solving ability when
they were exposed repeatedly with different
levels of temperature?

Within-Subject Factor
Name
• TEMP to denote the name of the within-
subject factor
• In the Number of Levels field, type 4 to
denote the four levels of the within-subject
factor TEMP (Temp1, Temp2, Temp3, and
Temp4)

Measure Name
• Type the name of the dependent measure:
ERRORS

Repeated as the contrast compares the mean of
each level (except the last) to the mean of the
subsequent level

Step 1
• If the Mauchly’s Test of Sphericity is not
significant (i.e., the assumption about the
characteristics of the variance-covariance
matrix is not violated), the Tests of Within-
Subjects Effects can be used.
• If the Mauchly’s Test of Sphericity is
significant, the Multivariate Tests should be
used.

Step 2
• As the within-subjects variable of TEMP is
statistically significant, results from the
Repeated contrast can be interpreted to
determine which variables contributed to the
overall difference.

Correlation
to describe the strength and
direction of the linear relationship
between two variables

Pearson’s correlation
• Is there any significant relationship between
time spent revising and exam performance?
• Is there any significant relationship between
anxiety and exam performance?

• Exam performance is positively related to the
amount of time spent revising, with a
coefficient of r = .397, which is also significant
at p < .001.
• Finally, exam anxiety appears to be negatively
related to the time spent revising, r = −.709, p
< .001.

• As the time spent revising increases, exam
performance will increase.
• As anxiety increases, exam performance will
decrease.

Data file
• Exam Anxiety.sav

Spearman’s correlation
• There is contest which require participants to
tell the biggest lie in the world
• There are 68 past contestants in this
competition
• They were placed in the competition (first,
second, third, etc.) and also gave them a
creativity questionnaire (maximum score 60).

Spearman’s correlation
• The position in the competition is an ordinal
variable because the places are categories but
have a meaningful order

Point–biserial correlations
• The relationship between the gender of a cat
and how much time it spent away from home

Regression
tell you how well a variable is /
variables are able to predict a
particular outcome

Simple regression
• Does advertising budget significantly predict
record sales?

• The value of R square is .335, which tells us
that advertising expenditure can account for
33.5% of the variation in record sales.

• The regression model overall predicts record
sales significantly well.

• Although this value is the slope of the
regression line, it is more useful to think of this
value as representing the change in the
outcome associated with a unit change in the
predictor.
• Therefore, if our predictor variable is increased
by one unit (if the advertising budget is
increased by 1), then our model predicts that
0.096 extra records will be sold.

Multiple Regression
• How well the Mastery scale and the Perceived
Control of Internal States Scale (PCOISS) are
able to predict scores on a measure of
perceived stress?
• OR
• Is there any significant contribution of all
predictor variables which are Mastery scale
and the Perceived Control of Internal States
Scale (PCOISS) towards variation in perceived
stress

R Square
• Tells you how much of the variance in the
dependent variable (perceived stress) is
explained by the model (which includes the
variables of Total Mastery and Total PCOISS).
• The value is .466 means that our model (which
includes Mastery and PCOISS) explains 46.6
per cent of the variance in perceived stress.

ANOVA
• The model in this example reaches statistical
significance (Sig. = .000; this really means
p<.0005).

• Of these two variables, mastery makes the
largest unique contribution (beta = -.42),
although PCOISS also made a statistically
significant contribution (beta = -.36)

Hierarchical Multiple Regression
• Flexible as it allows the researcher to
determine the order of entry of the
independent variables into the regression
equation.
• The order of entry is normally dictated by
logical or theoretical considerations.

Research Question
• If we control for the possible effect of age and
socially desirable responding, is this set of
variables (total mastery and PCOSISS) still able
to predict a significant amount of the variance
in perceived stress?

Research question
• Does noble values mediate the relationship
between knowledge about environmental
conservation and intention to practice
environmental conservation?

• H02: The relationship between knowledge
about environmental conservation and
intention to practice environmental
conservation will not be significantly mediated
by noble values.

Step 1
• Regressing the mediator on the independent
variable

Step 2
• Regressing the dependent variable on the
independent variable;

Step 3
• Regressing the dependent variable on both
the independent variable and on the
mediator.

To establish mediation, the following
conditions must hold:
• First, the independent variable must affect the
mediator in the first equation,
• Second, the independent variable must be
shown to affect the dependent variable in the
second equation,
• Third, the mediator must affect the
dependent variable in the third equation

• Complete mediation is indicated if the
previously significant independent variable
becomes non-significant in the final analysis.
• Partial mediation may occur when the
standardized regression coefficient of the
independent variable shows a reduction from
analyses of step 1 to step 3, but not to the
extent that it becomes non-significant.

History of moderation
• The technique comes out of ANOVA and
General Linear Model, in that multiple
regression is seen as an extension of analysis
of variance.
• Most people know of moderation from the
seminal Baron & Kenny article as well as a thin
paperback book by Leona Aiken and Stephen
West (1991) entitled Multiple regression:
Testing and interpreting interactions.

What is moderation?
• A moderating variable affects the
relationship of the IV on the DV.
• The moderator interacts with the IV to
predict outcome scores.
• Thus, certain levels of a moderator under
certain conditions of the IV might predict
different levels of the DV.

Hierarchical regression:
3 steps
DepressionAnxiety-C
Gender
Anxiety-C
X Gender
1st
step
2nd
step
3rd
step

• The model diagrammed in Figure 1 has three
causal paths that feed into the outcome
variable of task performance: the impact of
predictor (Path a), the impact of as a
moderator (Path b), and the interaction or
product of these two (Path c).

• The moderator hypothesis is supported if the
interaction (Path c) is significant.
• There may also be significant main effects for
the predictor and the moderator (Paths a and
b), but these are not directly relevant
conceptually to testing the moderator
hypothesis

• We speak of statistical interaction when a
relation between 2 variables (say X-Y) changes
as a function of a third variable (say Z).
• Note that the interactive effect is a
multiplicative effect, the effect of the product
oftwo IVs scores (i.e., XZ product).

Moderator
• It is a variable that changes the relationship
between an IV and a DV.
• A significant interaction between the
moderator and the IV means that the effect of
the IV on the DV changes depending on the
level of the moderator.
• We generally compare “high” levels of the
moderator (+1 standard deviation above the
mean) to “low” levels (-1 SD below the mean).

Mediators vs Moderators
• In mediation, the IV and the mediator are
associated (correlated), and the IV and the DV
are correlated, and there is an implied causal
path (“because”) that links the three variables.
• The IV causes the DV because the IV causes
the mediator which causes the DV.

Mediators vs Moderators
• In moderation (to get a significant
interaction), the IVs need not be correlated
with each other or with the DV.
• In moderation, the link between the IV and
the DV is different for high vs low levels of the
moderator. There is no because.
• It’s more like if-then contingencies: If there’s
high moderator, then the IV does this with the
DV, and if there’s low moderator, the IV does
this with the DV.

• The IV (self-esteem) impacts on grades (the
DV) but it’s moderated by motivation to study.
[At high motivation, there’s a link between
self-esteem and grades, but at low motivation,
there’s no link – everyone does badly.]

Right statistics for the right research questions

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