1. The document discusses parameter estimation, effect size, bivariate statistics including correlation and regression, and chi-square analysis. 2. Parameter estimation refers to using sample data to estimate population parameters, and sample statistics are estimations of population parameters. 3. Effect size measures the strength of the relationship between two variables and can be measured by eta square, partial eta square, and omega square, among others. 4. Correlation measures the association between variables, while regression predicts one variable from another. Chi-square analysis examines relationships between discrete variables.

1. Advance Research Method
Presented by
Muhammad Hannan Tahir
MBA 1.5 mor.
Parameter Estimation Effect Size
1 Bivariate Statistics: Correlation and Regression
2 Correlation
3 Regression
4 Chi-Square Analysis

2. Parameter Estimation
• The term parameter estimation refers to the process of using
sample data to estimate the parameters of the selected
distribution
• The population characteristic of interest is called a parameter.
• The corresponding sample characteristic is the sample statistic
or parameter estimate. Sample means are only estimations of
population means.
• sample means are unbiased estimators of population means,
the best guess about the size of a population mean (m) is the
mean of the sample randomly selected from that population.

3. Effect Size
• Effect size is a statistical concept that measures the strength of
the relationship between two variables on a numeric scale.
• For example: if we have data on the height of men and women
and we notice that, on average, men are taller than women, the
difference between the height of men and the height of women
is known as the effect size
• Effect size reflects the proportion of
variance in the DV that is associated
with levels of an IV.
It assesses the amount of total
variance in the DV that is predictable
from knowledge of the levels of the IV.

4. Effect Size cont..
• There are three primary effect size other then correlation and
regression coefficient, that are used when measuring association,
these are,
1 Eta square (η2) η2 =
𝑺𝑺 𝒆𝒇𝒇𝒆𝒄𝒕
𝑺𝑺 𝒕𝒐𝒕𝒂𝒍
• When there are two levels of the IV, η2 is the (squared) point
biserial correlation between the continuous variable (the DV)
and the dichotomous variable (the two levels of the IV).
• After finding a significant main effect η2 shows the proportion
of variance in the DV (SStotal) attributable to the effect
(SSeffect).
• Effect size measures how much association there is. A rough
estimate of effect size is available for any ANOVA through η2
(eta squared).

5. 2 Partial eta square (ηp2) ηp2 =
𝐒𝐒 𝐞𝐟𝐟𝐞𝐜𝐭
𝐒𝐒 𝐞𝐟𝐟𝐞𝐜𝐭+𝐒𝐒 𝐞𝐫𝐫𝐨𝐫
• An alternative form of η2 called partial η2 is available in
which the denominator contains only variance attributable to
the effect of interest plus error.
• This is also called strength of association.
• All effect size values are associated with the particular levels
of the IV used in the research and do not generalize to other
levels.
Effect Size cont..

6. Effect Size cont..
3 Omega square (ω2) ω2 =
𝐒𝐒 𝐞𝐟𝐟𝐞𝐜𝐭−(𝐝𝐟 𝐞𝐟𝐟𝐞𝐜𝐭)(𝐌𝐒 𝐞𝐫𝐫𝐨𝐫)
𝐒𝐒 𝐭𝐨𝐭𝐚𝐥+𝐌𝐒 𝐞𝐫𝐫𝐨𝐫
• A statistic developed to estimate effect size between IV and DV
in the population is ω2 (omega squared).
• where the denominator represents total variance, not just
variance due to effect plus error, and is limited to between-
subjects analysis of variance designs with equal sample sizes in
all cells
• Another type of effect size is Cohen’s d, basically a difference
between standardized means (i.e., means divided by their
common standard deviation).. Cohen (1988) has presented
some guidelines for small (η2 = .01), medium (η2 = .09), and
large (η2 = .25) effects.
4 Cohen’s d

7. Bivariate Statistics:
Correlation and Regression
• Correlation is the measure of the size and direction of the
linear relationship between the two variables, and squared
correlation is the measure of strength of association between
them.
• Correlation is used to measure the association between
variables; regression is used to predict one variable from
the other (or many others). However, the equations for
correlation and bivariate regression are very similar.

8. Correlation
• Correlation is used to measure the association between
variables
• The Pearson product- moment correlation coefficient, r, is
easily the most frequently used measure of association and the
basis of many multivariate calculations. The most interpretable
equation for Pearson r is
𝒓 =
𝒁 𝒙
𝒁 𝒚
𝐍 −𝟏
we can calculate Zx and Zy to calculate ZxZy
𝐙 𝐱 =
𝑿− 𝑿
𝐍 −𝟏
𝐳 𝐲 =
𝒚− 𝒚
𝐍 −𝟏
• Pearson r is independent of scale of measurement (because
both X and Y scores are converted to standard scores) and
independent of sample size (because of division by N)

9. Correlation cont..
• The value of r ranges between + 1 and – 1, where values close
to .00 represent no linear relationship or predictability between
the X and Y variables. An r value of + 1.00 or - 1.00 indicates
perfect predictability of one score when the other is known.
When correlation is perfect, scores for all subjects in the X
distribution have the same relative positions as corresponding
scores in the Y distribution.
𝒓 =
𝑵 𝒙𝒚 − ( 𝒙)( 𝒚)
[𝑵 𝒙 𝟐 − 𝒙) 𝟐 [ 𝑵 𝒚 𝟐 − ( 𝒚) 𝟐

10. Correlation cont…
• Positive correlation When both variables move in a same
direction
• Negative correlation Negative correlation
is a relationship between two variables in
which one variable increases as the other
decreases
• No correlation
If the correlation coefficient of two
variables is zero, it signifies that there is no
relationship between the variables

11. Regression
• Regression is used to predict a score on one variable from a
score on the other. In a bivariate regression where Y is
predicted from X, a straight line between the two variables is
found. The best- fitting straight line goes through the means of
X and Y and minimizes the sum of the squared distances
between the data points and the line.
To find the best- fitting straight line, an equation is solved of
the form
Y= A + BX
where Y_ is the predicted score, A is the value of Y when X is
0.00, B is the slope of the line (change in Y divided by change in
X), and X is the value from which Y is to be predicted.

12. Regression cont…
• The bivariate regression coefficient, B, is the ratio of the
covariance of the variables and the variance of the one from
which predictions are made.
𝑩 =
𝑵 𝒙𝒚 − ( 𝒙)( 𝒚)
𝑵 𝒙 𝟐 − ( 𝒙) 𝟐
• In regression, the variance of the predictor variable serves as
the denominator; if Y is predicted from X, X variance is the
denominator, whereas if X is predicted from Y, Y variance is
the denominator. To complete the solution, the value of the
intercept, A, is also calculated.
• The difference between the predicted and the observed
values of Y at each value of X represents errors of
prediction or residuals. The best- fitting straight line is the
line that minimizes the squared errors of prediction.

13. Chi-Square Analysis
• the chi- square (𝑥2) test of independence is used to examine
the relationship between two discrete variables. If, for
instance, one wants to examine a potential relationship
between region of the country (Northeast, Southeast, Midwest,
South, and West) and approval versus disapproval of current
political leadership, 𝑥2 is the appropriate analysis.
• In 𝑥2 analysis, the null hypothesis generates expected
frequencies against which observed frequencies are tested. If
the observed frequencies are similar to the expected
frequencies, then the value of 𝑥2
is small and the null
hypothesis is retained; if they are sufficiently different, then
the value of 𝑥2
is large and the null hypothesis is rejected.

14. Chi-Square Analysis
• The relationship between the size of 𝑥2 and the difference in
observed and expected frequencies can be seen readily from
the following computational equation for 𝑥2:
(𝒇𝒐−𝐅𝐞) 𝟐
𝑭𝒆
• where fo represents observed frequencies
and Fe represents the expected frequencies
in each cell. Summation is over all the cells
in a two- way table.
• Usually, the expected frequencies for a cell
are generated from the sum of its row and
the sum of the column.

15. Chi-Square Analysis
𝐂𝐞𝐥𝐥 𝐅𝐞 =
𝐫𝐨𝐰 𝐬𝐮𝐦 ( 𝐜𝐨𝐥𝐮𝐦 𝐬𝐮𝐦)
𝐍
• When this procedure is used to generate the expected
frequencies, the null hypothesis tested is that the variable on
the row is independent of the variable on the column. If the fit
to the observed frequencies is good then one concludes that the
two variables are independent; a poor fit leads to a large x2,
• rejection of the null hypothesis, and the conclusion that the
two variables are related.