Inferential Statistics
ARIEL F. MELAD
2020 (EDITED)
Learning Objectives
At the end of the module, you should be able to:
review types of
measurements
variable
and
different
level
of
differentiate between parametric and non-parametric test
choose appropriate statistical tool needed for test of
difference or test of relationships
perform basic statistical analysis using statistical software
interpret and report the statistical results of the analysis
Topics
Summary Statistics
Appropriate Statistical Methods
Most common statistical test
Independent samples t-test
Paired t-test
One-way Analysis of Variance (ANOVA)
Chi-square test
Pearson correlation
Summary Statistics
Summary Statistics
Summary of descriptive and graphical statistics
Chart
Pie chart or bar
chart
Variable type
Purpose
Summary Statistics
One categorical Shows frequencies/percentages/
proportions
Class percentages
Stacked/multiple Two categorical Compares proportions within
bar
groups
Histogram
One scale
Show distribution of results
Scatter graph
Two scale
Boxplot
One scale/one
categorical
Scale by time
Line chart
Shows relationship between two
variables
Compare spread of values
Display changes over time
Comparison of groups
Percentages with groups
Mean and standard
deviation
Correlation coefficient
Median and IQR
Means by time point
Inferential Statistics
Appropriate Statistical Methods
Things to consider:
1. which is dependent and independent variables?
2. What type of measurement scales/data?
3. How many variables are there?
4. Is it test of difference or relationship?
Inferential Statistics
Common single comparison test
Comparing
Average of two
INDEPENDENT groups
Average of 3+
independent groups
The average difference
bewteen paired
(matched) samples e.g.
weight before and after
The 3+ measurements on
same subject
Dependent
(outcome)
variable
Scale
Scale
Scale
Scale
Independent
(explanatory)
variable
Parametric test Non-parametric test
(data is
(ordinal/skewed
normally
data)
distributed)
Independent
Man –Whitney
Nominal (Binary) samples t-test test/Wilcoxon rank
sum
Nominal
One-way
Kruskal-Wallis test
ANOVA
Time/condtion
Paired t-test Wilcoxon signed rank
varaible
test
Time/condtion
varaible
Repeated
measures
ANOVA
Friedman test
Test of Association
Dependent
(outcome)
variable
Independent
(explanatory)
variable
Parametric test
(data is normally
distributed)
Non-parametric
test
(ordinal/
skewed data)
Relationship between 2
continuous variables
Scale
Scale
Transform the
data
Predicting the value of one
variable from the value of a
predictor variable or looking
for significant relationship
Scale
Any
Pearson’s
correlation
coefficient
Simple linear
regression
Nominal
(binary)
Categorical
Any
Comparing
Assesing the relationship
between two categorical
variable
Categorical
Logistic regression
Chi-square
test
Inferential Statistics
Statistical Significance
Statisticians often choose a cut-off point under which a pvalue must fall for a finding to be considered statistically
significant.
If the p-value is less than or equal 0.05 (5%), the result is
deemed statistically significant, i.e., there is a significant
relationship between the variables. Use the p-value as an
indicator of statistical significance.
Inferential Statistics
Statistical Significance
Two forms of inferential statistics exist: test that examine
associations, and test that examine differences.
Most common statistical test
1. Descriptive Analytics
uses percentages, frequency counts, means, ranks and
weighted mean.
Most common statistical test
Example1. Let us examine data on SPSS
Open the file, “ Weight Management”.
Q1: What is the profile of the participants in terms of sex, age,
initial BMI (interpretation) and final BMI (interpretation) of the
participants.
Variables: sex, age, initial and final BMI (interpretation).
Types of data: all variable are categorical.
Descriptive Analytics
Interpretation
Table shows the profile of participants in terms of sex. It can
be seen on the table that 18 or 46.2% are male while 21 or
53.8% are female. This shows that most of the participants in
the weight management activity are female.
Descriptive Analytics
Interpretation
According to Craft, Carroll and Lustyk (2014) on their study
entitled “ Gender Differences in Exercise Habits and Quality of
Life Reports”, results revealed that women reportedly
significant higher exercise and quality of life levels than men.
Women reported exercising for weight loss and toning more
than men, whereas men reported exercising for enjoyment
more than women.
Descriptive Analytics
Table shows the initial weight of the participants before the
weight management activity. It can be viewed from the table that
21 or 53.8% have a normal weight, 11 or 28.2% are overweight
while only 7 or 17.9% are obese. This shows that most of the
participants have normal weight before the activity conducted.
Descriptive Analytics
According to U.S National Library of Medicine, regular exercise
and physical activity may help you to control your weight,
reduce your risk of heart diseases, help your body manage
blood sugar and insulin levels, improve your mental health and
mood, reduce your risks of some cancers and many more.
Descriptive Analytics
Can you interpret?
𝑥 age is 23.10 years old
Descriptive Analytics
Can you interpret this data?
Most common statistical test
1. Independent samples t-test
Independent variable: categorical (dichotomous)
Dependent variable: scale/continuous
Use: A t-test is used to compare the means between
two independent groups or categories.
Plot: Histogram of differences
Important: the dependent variable must be normally distributed.
Independent Samples T-test
Interpretation:
If the p-value < 0.05, there is a significant difference
between the means of the two groups. Report the means of
the two groups or the mean difference and confidence
interval from the SPSS output to describe the difference.
Independent Samples T-test
Example1. Let us examine data on SPSS
Open the file, “ Weight Management”.
Q1: Is there a significant difference between the weight
goal of the male and female participants before the weight
management activity?
Variables: sex and weight goal
Independent variable: sex
Independent Samples T-test
dependent variable: weight goal
Sex is categorical with only 2 groups (dichotomous)
Weight goal is scale/ numerical data.
Independent Samples T-test
Independent Samples T-test
Sample write-up:
“An independent samples t-test was conducted to examine
whether there was a significant difference between male and
female group in relation to their weight goal before the conduct
of the weight management activity. The test revealed a
statistically significant difference between male and female
group since the p-value is < 0.05 level of significance(t = 2.058,
df = 36, p = .047).
Independent Samples T-test
Sample write-up:
Male group (mean = 61.67, SD = 6.86) reported
significantly higher weight goal than the female
group (mean = 56.67, SD = 8.11)”. This can be
concluded that male are more confident on weight
loss before the weight management activity than the
female group.
Independent Samples T-test
Sample write-up:
For men, exercise itself was the best predictor of
quality of life. In other words, higher levels of
exercise in men were associated with higher quality
of life. This suggests that men may be able to
improve their quality of life with increased exercise,
no matter what reasons for exercise men give(Craft,
Carroll & Lustyk ,2014) .
One way ANOVA (Analysis of Variance)
Independent variable: categorical (at least three categories)
Dependent variable: scale/continuous
Use: one way ANOVA is used to detect the difference in
means of 3 or more independent groups. ANOVA is the
generalize version of independent samples t-test.
Plot: Box-plots or confidence interval plots
Important: the dependent variable must be normally
distributed.
One way ANOVA
Note:
Post-hoc tests allow you to determine where significant
differences lie.
When the ANOVA is found to be significant, one must
examine which two groups differ significantly from the
total number of groups: on post-hoc tests table, look at
mean differences between different pairs.
One way ANOVA
Note:
There are many post-hoc tests to choose from when doing
an ANOVA.
The Scheffe post-hoc test should be selected when equal
variances assumed but the Games- Howell post-hoc test
should be selected if not.
One way ANOVA
Example2. Let us examine data on SPSS
Open the file, “ Weight Management”.
Q1: Is the weight loss among participants differs on the
frequency of exercise?
Variables: weight loss and frequency of exercise
Independent variable: frequency of exercise
One way ANOVA
dependent variable: weight loss
Frequency of exercise is categorical with 3 groups (3x,4x
and 5x a week)
Weight loss is scale/ numerical data.
One way ANOVA
One way ANOVA
Sample write-up:
“A one-way ANOVA was conducted to examine whether
there were statistically significant differences among
participants in different frequency of exercise in relation to
their weight loss after the weight management activity. The
results revealed statistically no significant difference on
their weight loss as to their frequency of exercise since the
p-value is > 0.05 level of significance, F = 0.146, p = 0.864.
Paired T-test
Use: Paired t-test is used to compare significant difference
two population means
Important: Variables between two groups must be scale.
Example: Before-and-after observations on the same
subjects (e.g. blood pressure measurements before and
after exercise)
Paired T-test
Example3. Let us examine data on SPSS
Open the file, “ Weight Management”.
Q1: Is there a significant difference on the BMI of the
participants before and after the weight management
activity?
Group 1: BMI before (scale variable)
Group 2. BMI after (scale variable)
Paired T-test
Paired T-Test
Sample write-up:
“A paired t-test was conducted to examine whether there
were statistically significant differences on the BMI of the
participants before and after the weight management
activity. The results revealed statistically significant
difference on the BMI of the participants before and after
the activity since the p-value is < 0.05 level of significance
(t=18.09, df =38, p = 0.000).
Paired T-Test
Sample write-up:
This shows that the reported mean BMI of the
participants before (mean = 24.9, SD = 3.65) the activity is
higher than after (mean = 23.75, SD = 3.62) the weight
management exercise. This further shows that there was a
decrease of an average of 1.20 kg on the BMI score of the
participants after the weight management activity. Hence,
the weight management activity is effective.
Chi-square Test
Use: The Pearson Chi square is used to test whether a
statistically significant relationship exists between two
categorical variables. It accompanies a crosstabulation
between the two variables.
Independent variable: categorical
Dependent variable: categorical
Chi-square Test
Example: educational attainment and use of contraceptives
Independent variable: educational attainment (e.g. Elem
grad, HS grad, College Grad, etc…)
Dependent variable: use of contraceptives (e.g. Yes or No)
Independent variable: categorical data
Dependent variable: categorical data
Chi-square Test
Example4. Let us examine data on SPSS
Open the file, “ Weight Management”.
Q1: Is there a significant relationship between the initial
weight and initial BMI of the participants?
Independent: initial weight
Dependent: initial BMI
 initial weight is categorical (e.g. 20-30, 30-40, etc.)
initial BMI is categorical (e.g. 10-15, 16-20, etc.)
Chi-square Test
Chi-square Test
Sample write-up:
“A Pearson chi-square test was conducted to examine
whether there was a relationship between initial weight and
initial BMI of the participants. The results revealed that
there was a significant relationship between the two
variables (Chi square value = 67.10, df =20, p=0.000) since
the p-value <0.05 level of significance. This means that the
higher your initial weight the higher your BMI is. (see table
1 and 2)
Pearson Correlation
Use: Correlation tests (Pearson correlation) are used
to examine relationships between two or more
quantitative/numerical variables.
They measure the strength and direction of a
relationship between variables.
Pearson Correlation
Pearson Correlation
It ranges from negative (-1) to positive (+1)
coefficient values.
A negative correlation indicates that high values on
one variable are associated with low values on the
next. A positive correlation indicates that high values
on the one variable are associated with high values
of the next.
Pearson Correlation
Example 1:
A positive correlation between salary and
expenditures means that higher your salary the
higher your expenses is.
Pearson Correlation
Example 2:
A negative correlation between the number of
absences and score during exams means that the
more absences you take place in
the class the lower your score is
during exams.
Pearson Correlation
Example 3:
No correlation occurred between
your height and your expenditures.
Pearson Correlation
The p-values tells you whether the relationship or
correlation between the variables are statistically
significant (p< .05).
Pearson Correlation
Strength
.10 to .29 – weak relationship
.30 to .49 – moderate relationship
.50 and above – strong relationship
Pearson Correlation
The sign of the relationship does not indicate the
strength; (-).50 is the same strength as (+).50 but
different direction.
‘r’is the symbol of the correlation coefficient.
Pearson Correlation
Example5. Let us examine data on SPSS
Open the file, “ Weight Management”.
Q1: Is there a significant relationship between the
weight and BMI of the participants?
Independent: weight
Dependent: BMI
Both variables are numerical/quantitative
Pearson Correlation
Pearson Correlation
Sample write-up
“A Pearson correlation analysis was conducted to examine
whether there is a relationship between weight and BMI.
The results revealed a significant and strong positive
relationship (r = .913, p = .000). The higher your weight
score the higher your BMI (see Table 1).”
Pearson Correlation
Sample write-up
“The result of the present study is similar to the study of
Islam et al (2017) showed that weight and height among
males and females is significantly correlated.”
Pearson Correlation
Try this!
Is weight significantly
correlated with expenditures?
Comment
below
References
Davonish, D. (n.d). Exploring relationships using SPSS
inferential statistics (part II).