The document discusses bivariate and multivariate linear regression analysis, explaining how to estimate regression coefficients using software like SPSS and interpret their results. It covers topics such as estimating and interpreting intercept and slope coefficients, measuring predictive power using R-squared, and testing the significance of individual regression coefficients and the overall regression model through techniques like t-tests and F-tests.
Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables.
Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables.
Regression analysis is a powerful statistical method that allows you to examine the relationship between two or more variables of interest. Regression analysis is a reliable method of identifying which variables have impact on a topic of interest. The process of performing a regression allows you to confidently determine which factors matter most, which factors can be ignored, and how these factors influence each other.In this presentation a brief introduction about SLR and MLR and their codes in R are described
In statistics, regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'Criterion Variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function which can be described by a probability distribution.
This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor.
For more topics stay tuned with Learnbay.
Deep in the recesses of the brain lies the most ancient of all our faculties: The Lizard brain; It’s a mysterious place of snap judgements and life-saving instincts. Design can reach it, but first let’s understand it, and maybe get to know ourselves and our audience along the way.
Regression analysis is a powerful statistical method that allows you to examine the relationship between two or more variables of interest. Regression analysis is a reliable method of identifying which variables have impact on a topic of interest. The process of performing a regression allows you to confidently determine which factors matter most, which factors can be ignored, and how these factors influence each other.In this presentation a brief introduction about SLR and MLR and their codes in R are described
In statistics, regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'Criterion Variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function which can be described by a probability distribution.
This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor.
For more topics stay tuned with Learnbay.
Deep in the recesses of the brain lies the most ancient of all our faculties: The Lizard brain; It’s a mysterious place of snap judgements and life-saving instincts. Design can reach it, but first let’s understand it, and maybe get to know ourselves and our audience along the way.
Multivariate data analysis regression, cluster and factor analysis on spssAditya Banerjee
Using multiple techniques to analyse data on SPSS. A basic software that can easily help run the numbers. Multivariate Data Analysis runs regressions models, factor analyses, and clustering models apart from many more
Introduces and explains the use of multiple linear regression, a multivariate correlational statistical technique. For more info, see the lecture page at http://goo.gl/CeBsv. See also the slides for the MLR II lecture http://www.slideshare.net/jtneill/multiple-linear-regression-ii
Lecture 4 - Linear Regression, a lecture in subject module Statistical & Mach...Maninda Edirisooriya
Simplest Machine Learning algorithm or one of the most fundamental Statistical Learning technique is Linear Regression. This was one of the lectures of a full course I taught in University of Moratuwa, Sri Lanka on 2023 second half of the year.
For this assignment, use the aschooltest.sav dataset.The dMerrileeDelvalle969
For this assignment, use the aschooltest.sav dataset.
The dataset consists of Reading, Writing, Math, Science, and Social Studies test scores for 200 students. Demographic data include gender, race, SES, school type, and program type.
Instructions:
Work with the aschooltest.sav datafile and respond to the following questions in a few sentences. Please submit your SPSS output either in your assignment or separately.
1. Identify an Independent and Dependent Variable (of your choice) and develop a hypothesis about what you expect to find. (
note: the IV is a grouping variable, which means it needs to have more than 2 categories and the DV is continuous)
2. Run Assumption tests for Normality and initial Homogeneity of Variance. What are your results?
3. Run the one-way ANOVA with the Levene test & Tukey post hoc test.
a. What are the results of the Levene test? What does this mean?
b. What are the results of the one-way ANOVA (use notation)? What does it mean?
c. Are post hoc tests necessary? If so, what are the results of those analyses?
4. How do your analyses address your hypotheses?
Is concentration of single parent families associated with reading scores?
Using the AECF state data, the regression below measures the effect of the state's percentage of single parent families on the percentage of 4th graders with below basic reading scores.
%belowbasicread = β0 + β1x%SPF + u
Stata Output
1) Please write out the regression equation using the coefficients in the table
2) Please provide an interpretation of the coefficient for SPF
3) How does the model fit?
4) What is the NULL hypothesis for a T test about a regression coefficient?
5) What is the ALTERNATE hypothesis for a T test about a regression coefficient?
6) Look at the p value for the coefficient SPF.
a) Report the p value
b) How many stars would it get if we used our standard convention?
* p ≤ .1 ** p ≤ .05 *** p ≤ .01
image1.png
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Tw ...
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very detailed illustration of Log of Odds, Logit/ logistic regression and their types from binary logit, ordered logit to multinomial logit and also with their assumptions.
Thanks, for your time, if you enjoyed this short article there are tons of topics in advanced analytics, data science, and machine learning available in my medium repo. https://medium.com/@bobrupakroy
Regression Analysis is simplified in this presentation. Starting with simple linear to multiple regression analysis, it covers all the statistics and interpretation of various diagnostic plots. Besides, how to verify regression assumptions and some advance concepts of choosing best models makes the slides more useful SAS program codes of two examples are also included.
1. Statistical Analysis Software
Click to edit Master title style
Bivariate and Multivariate Regression
Analysis
Academic Department of Marketing
Caucasus School of Business
Caucasus University 1
2011
2. Problems of Test 1
• Formulating null and alternative
hypotheses incorrectly
• Ignoring question “why”
• Ignoring the necessity to comment on
the scale used
• Mixing up Wilcoxon and paired
samples T test
• Massively ignoring the necessity to
check the equality of variances
(Levene’s test)
• Kolmogorov-Smirnov test
3. Homework 1
• Three or four homework assignments will be
given throughout the course. You will be
informed about the number of points you can
get from each assignment.
• The first homework assignment will include
two problems: the first one is the ANOVA
problem from test 1 – each one of you will
have individual databases. The second problem
will be about using Pearson’s Chi Square
statistic in cross-tabulations. However, you will
have to come up with your own example and
your own fictional database.
• The assignment is worth 2 points and is due
4. Important Note (Homework)
• EVEN IF ALL THE INTERPRETATION IS
CORRECT, YOU WILL GET ZERO POINTS
IN CASE YOU SUBMIT THE WRONG
OUTPUT WHETHER IT’S BECAUSE YOU
DID THE WRONG TEST OR YOU USED
SOMEBODY ELSE’S DATASET.
5. Warming Up – Linear Equations
• What does a linear relationship imply?
• How does a linear relationship look like
(mathematically)?
• What are the variables in this equation
and what are the parameters?
• How are the parameters interpreted?
6. Scatterplot (1)
• Scatterplot – collection of points (x,y) on the
coordinate system. Each point on a scatterplot
depicts a single case, that has a specific X value
and a specific Y value, which you can find on the
X and Y axis.
7. Scatterplot (2)
• As we see, there is a certain relationship
between income and saving – the higher
the income, the higher the saving.
• But are we interested only in the
direction? Not really. It is important to
measure by how much saving increases as
income increases by, say, 1 Lari.
• By saying this we imply that there is a
linear relationship between income and
saving (which is not necessarily true, but
let’s ignore this for now).
8. Scatterplot (3)
• Going back to our scatterplot, we need
to find a line (i.e. determine the
intercept and the slope) which best
describes the relationship between two
variables (in this case saving and
income).
• This is exactly where regression comes
into play – it helps to identify such a line
by using the sample information.
9. Bivariate Regression Model
• In theory, the relationship between saving
and income already exists and is
somewhere out there – we can’t really
determine it in practice. Why? Because we
would need to collect information about
everybody’s income and everybody’s
saving (i.e. we would need information
about the whole population).
• If we could, the bivariate regression model
would look like this:
Y=β0+ β1*X, where Y is saving and X is
income.
10. Error Term
• Note that even in the ideal case, where we
have information about the population, we are
still unable to exactly predict the level of saving
by the level of income. Why? Because income
is not the only factor that determines saving.
There are other factors that aren’t accounted
for in our bivariate regression model.
• All the other factors not explicitly accounted
for in the regression model fall in the so called
error term, denoted by ε.
• Therefore, the population regression model
looks like this:
Y=β0+ β1*X+ ε
11. Linear Regression Analysis
(Bivariate)
• Identifying the line that depicts the
relationship between X and Y boils down to
estimating β0 and β1.
• What a regression does is basically
providing us with estimates (regression
coefficients) of β0 and β1, which are
denoted by b0 and b1.
• The estimated regression model looks like
this:
Ŷ= b0 + b1*X
12. Interpreting Regression
Coefficients
• Ŷ= b0 + b1*X
• Ŷ – predicted values, shows us the predicted
values of Y as X takes specific values.
• b0 - intercept, shows the predicted value of Y
when X=0.
• b1 - slope estimate, shows by how much the
predicted value of Y changes as X changes by 1
unit.
13. Residual
• Residual is the difference between the
actual value of Y and predicted value of
Y, and is denoted by e.
• e=Y – Ŷ
• Do not mix up residual and error term.
They are NOT the same. We never know
the error term. However, we can easily
estimate the residual. Residual is an
estimate of the error term.
15. Linear Regression - Output
• Thus, if income is 0, the predicted saving is
equal to 124.842. And if income increases by
1 Lari, saving will increase by 0.147 Lari.
• Is this model appropriate to predict the levels
of saving? Not really. Saving is also
determined by other factors, like family
size, education level of household
head, his/her age and gender. (Of course
there may be other determinants as well, but
let’s focus on these for now)
16. Multiple Regression Analysis
• Multiple regression implies including more
than one independent variable in the
regression model. Basically it looks like this:
Y=β0+ β1*X1+ β2*X2+ β3*X3+…+ βk*Xk+ ε
• In this case we need to estimate (k+1)
parameters - b0, b1, b2 … bk.
• Interpretation of slope coefficients: b1
shows by how much predicted Y changes as
X1 changes, holding all other X-s constant.
• Interpretation of intercept – the predicted
value of Y when all the X-s are equal to
zero.
18. Major Goals of Conducting
Regression Analysis
• Goal 1. Measuring partial effects – by how
much does Y change when X1 changes by 1
unit, holding all other X-s constant?
• Goal 2. Forecasting the values of the
dependent variable – what is the predicted
saving level (measured in Laris) of a family
that has a family income of 1000 Laris, that
has 5 members, whose household head
studied for 15 years and whose household
head is 47 years old?
• Regression provides answers to these
questions.
19. Predictive Power of a Model
• In order to know how good our model
is for forecasting, we need to measure
the predictive power of the model. In
other words, we want to know how
well the independent variables explain
the dependent variable.
• Coefficient of determination (R-
squared) is widely used for this
purpose.
20. Coefficient of Determination –
R-Squared (1)
• Coefficient of determination (R-squared)
measures the portion of the variation in Y
explained by the variation in X-s, in other
words, how much of the variation in the
dependent variable is explained by the
independent variables.
• This is also called goodness-of-fit.
• R-squared ranges from 0 to 1 and shows how well
the regression line describes the data cloud that
you see on the scatterplot.
• The closer the data are clustered around the
regression line, the closer the R-squared is to 1.
R2=1 is perfect fit (never possible in practice). The
closer the R-squared is to 0, the worse the fit.
21. Coefficient of Determination –
R-Squared (2)
• For example, if R-squared is equal to
0.045, it means that independent
variables explain only 4.5% of variation in
the dependent variable.
• This is an example of low predicting
power.
• The higher the R-squared, the better the
predictive power of your model.
22. Testing Significance of Regression
Coefficients (1)
• As we already mentioned, the other goal
of regression analysis is to determine
partial effects.
• Basically, partial effects measure pure
effects of respective independent
variables on the dependent variable.
• What we want to know is whether these
pure effects are important. How can we
find this out?
• This is done by testing the significance of
the regression coefficients.
23. Testing Significance of Regression
Coefficients (2)
• Suppose we want to test whether age
of household head (X4) has an
important effect on saving once all the
other factors (household
size, income, education of household
head) are controlled for.
• Null hypothesis is that β4 = 0. (i.e., as X4
changes by 1 unit, nothing happens to
Y, no effect on Y)
• Alternative hypothesis is that β4 is
different from 0 (two-tailed test).
24. Testing Significance of Regression
Coefficients (3)
• It can be shown that if we divide the estimate of
β4 (b4) by standard error of b4 (which is standard
deviation of b4 ), the resulting statistic follows t
distribution.
• Thus, we can either calculate the t statistic and
compare it to the critical t value at 5%
significance level, or we can simply look at the p-
value (Sig.) of the regression coefficient. If the
latter is less than 0.05, we conclude that the
regression coefficient is significantly different
from zero (or just significant, shortly). In other
words, the partial effect of this variable is
statistically important.
25. Testing Significance of Regression
Coefficients - Example
• Going back to our multivariate regression
example, no single independent variable
appears to be statistically significant – all
the p-values are more than 0.05.
• However, even though these variables
are separately insignificant, there is a
chance that they are collectively
significant.
• This hypothesis is tested by joint F test.
26. Joint F Test
• Null Hypothesis: β1 = β2 = β3 = β4 = 0
• Alternative Hypothesis: at least one of them is different
from zero.
• This is equivalent to testing whether R2=0.
27. Important Note
• It can happen that all the coefficients
are separately insignificant but jointly
significant, even though in our example
they’re also jointly insignificant at 5%
significance level.
• It can also happen that regression
coefficients are separately significant
but jointly insignificant. WHEN?