This document discusses multiple linear regression analysis conducted to assess staff satisfaction levels at an educational institution. A questionnaire was administered to staff across multiple locations. Factor analysis was used to identify the variables that best predict overall satisfaction. A regression model was developed using satisfaction as the dependent variable and questions regarding workplace expectations, resources, communication, recognition, development opportunities, and opinions as independent variables. The model was analyzed in SPSS and showed high explanatory power, with no issues of multicollinearity between predictors.
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 defines basics of regression analysis for students and scholars. uses, objectives, types of regression, use of spss for regression and various tools available in the market to calculate regression analysis
Simple Linear Regression: Step-By-StepDan Wellisch
This presentation was made to our meetup group found here.: https://www.meetup.com/Chicago-Technology-For-Value-Based-Healthcare-Meetup/ on 9/26/2017. Our group is focused on technology applied to healthcare in order to create better healthcare.
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
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.
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 defines basics of regression analysis for students and scholars. uses, objectives, types of regression, use of spss for regression and various tools available in the market to calculate regression analysis
Simple Linear Regression: Step-By-StepDan Wellisch
This presentation was made to our meetup group found here.: https://www.meetup.com/Chicago-Technology-For-Value-Based-Healthcare-Meetup/ on 9/26/2017. Our group is focused on technology applied to healthcare in order to create better healthcare.
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
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.
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
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This paper proposes a novel model management technique to be applied in population- based heuristic optimization. This technique adaptively selects different computational models (both physics-based and statistical models) to be used during optimization, with the overall goal to end with high fidelity solutions in a reasonable time period. For example, in optimizing an aircraft wing to obtain maximum lift-to-drag ratio, one can use low-fidelity models such as given by the vortex lattice method, or a high-fidelity finite volume model (that solves the full Navier-Stokes equations), or a surrogate model that substitutes the high-fidelity model.The information from models with different levels of fidelity is inte- grated into the heuristic optimization process using a novel model-switching metric. In this context, models could be surrogate models, low-fidelity physics-based analytical mod- els, and medium-to-high fidelity computational models (based on grid density). The model switching technique replaces the current model with the next higher fidelity model, when a stochastic switching criterion is met at a given iteration during the optimization process. The switching criteria is based on whether the uncertainty associated with the current model output dominates the latest improvement of the fitness function. In the case of the physics-based models, the uncertainty in their output is quantified through an inverse assessment process by comparing with high-fidelity model responses or experimental data (if available). To determine the fidelity of surrogate models, the Predictive Estimation of Model Fidelity (PEMF) method is applied. The effectiveness of the proposed method is demonstrated by applying it to airfoil optimization with the objective to maximize the lift to drag ratio of the wing under different flow regimes. It was found that the tuned low fidelity model dominates the optimization process in terms of computational time and function calls.
A researcher in attempting to run a regression model noticed a neg.docxevonnehoggarth79783
A researcher in attempting to run a regression model noticed a negative beta sign for an explanatory variable when s/he was expecting a positive sign based on theoretical considerations. What advice would you give to the researcher as to what is going on and what specific diagnostics would you look at? Explain conceptually and statisticallythe different ways you cancorrect for this problem.
Reason
One of the most common and important reasons for such situations is the existence of multicollinearity. Multicollinearity can happen if some of the independent variables are highly correlated to each other or to another variable that is not in the model.
Multicollinearity also has other symptoms such as
· Large variance for regression coefficients
· Non-significant individual coefficients while the general model is significant
· Change of marginal contributions depending on the variables in the model
· Large correlation coefficients in the correlation matrix of variables
It should however be noted that the general model can preserve its predictive ability and it is only the explanatory power that is lost
Before going to the solutions and measures the researcher can take it is wise to take a step back and see the underlying reason for the multicollinearity. An extreme case where two variables are identical gives the best understanding of problem
In this case we are trying to define y as a function of and while in reality . Therefore any linear combination of and is replaceable by infinite other linear combinations (ie )
It is simply understandable that while the y is predicted correctly in all the instances individual coefficients for and are meaningless.
Diagnosis
One of the most common diagnoses for multicollinearity is the variance inflation factor (VIF)
Where
And is the coefficient of multiple determination of regression of on other variables
The variance inflation factor therefore determines how much the variance of each coefficient inflates. when equals zero VIF equals 1 which suggests zero multicollinearity heuristic is that any value of VIF larger than 10 is alerting and a case of strong multicollinearity exists.
Solution
s
There are a few solutions for the multi Collinearity problem:
1- Ignoring the problem completely is possible for cases where we only care about the final model fit and prediction capability rather than individual coefficients and explanation power
2- Removing some of the correlated variables from the model, this can be justified since we can argue the effect of variable is however seen by similar highly correlated variables that are kept in the model
3- Principle component analysis (or any orthogonal transformation) can reduce the number of factors to a few orthogonal factors with no collinearity; however we should note that the interpretation of variables after a PC transformation is hard.
4- For cases where we intend to keep all the variables in the model without any major transformation, the Ridge regr.
The objective of the project is to use the dataset 'Factor-Hair-Revised.csv' to build an optimum regression model to predict satisfaction.
Perform exploratory data analysis on the dataset. Showcase some charts, graphs. Check for outliers and missing values.
Check evidence of multicollinearity.
Perform simple linear regression for the dependent variable with every independent variable.
Perform PCA/Factor analysis by extracting 4 factors. Interpret the output and name the Factors.
Perform Multiple linear regression with customer satisfaction as dependent variables and the four factors as independent variables.
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
1. • Many applications of regression
analysis involve situations in which
there are more than one regressor
variables.
• A regression model that contains more
than one regressor variables is called a
multiple regression model.
Introduction
Multiple Linear Regression Model
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make Project Reports or using SPSS to evaluate results
2. A study conducted to assess the satisfaction
levels of staff from an educational institution
with branches in a number of locations across
the country.
Staff were asked to complete a short,
anonymous questionnaire (shown later in this
Appendix) containing questions about their
opinion of various aspects of the organization
and the treatment they have received as
employees.
Multiple Linear Regression Model
Problem Statement
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
3. • There are many variables exists in the research
therefore factor analysis has been conducted to
make decision which variables are best fit for the
model by Extraction Method.
• The factor analysis extracted Q1,Q2,Q3 but when
Including the variables Q4,Q5and Q6 also gave the
right results
Factor Analysis of Variable to fit into the model
Multiple Linear Regression Model
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
4. The error term has a normal distribution
with a mean of 0.
The variance of the error term is constant
across cases and independent of the
variables in the model.
There is no multicolinearity
Multiple Linear Regression
Model
Assumptions:
Rizwan Manzoor (MSc Finance)
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5. Multiple Linear Regression Model
• For example, suppose that the Satisfaction level of staff
is depend on the Q1,Q2,Q3,Q4,Q5 and Q6. A possible
multiple regression model could be
Model
Y=β0+ β1Q1+ β2Q2+ β3Q3 +β4Q4+β5Q5+β6Q6+E
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
6. Multiple Linear Regression Model
where
Y - Total Staff Satisfaction Scale (Dependent)
Q1 - Is it clear what is expected of you at work? (Independent)
Q2 - At work have you been provided with all the equipment and materials
required for you do your work efficiently? (Independent)
Q3 - Does the organization keep you up to date with information concerning
development and changes? (Independent)
Q4 - Do you receive recognition from the organization for doing good Work ?(Independent)
Q5 - Does your manager or supervisor encourage your development at work?(Independent)
Q6 - Do you feel that your opinions seem to count to the organization? (Independent)
E - Error Term (Influence of other Variables/factors)
Introduction
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
8. Multiple Linear Regression Model
Example: Table 1.2
Education Institution
Staff Survey
conducted in across
Australia
Example. 536
Observations has
been taken to check
the reliability of the
Multiple Linear
Regression Model.
By taking Staff Total Satisfaction as dependent
variable and Q1,Q2,Q3,Q4,Q5 and Q6 as
Independent Variables
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
9. Multiple Linear Regression Model
For running the Multiple Linear Regression
and testing the result in SPSS the variables
has been set according their dependency and
independency. By Using Enter Method
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
10. Multiple Linear Regression Model
Interpreting Regression Output
•The Coefficient of determination or adjusted R2 can be interpreted because
of more than one independent variables.
•This is the percentage(%) of total variation in the dependent variable due to
the independent variable.
•We can see that Adjusted R square value 0.920 that our independent
variables explain 92.% of the variability of our dependent variable.
•Std. Error of the estimate shows that the other variables, that have not been
taken in the model, have influence 1.995%. The std Error is low due the
number of variables. The larger the size of variables the smaller the std.Error.Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
11. Multiple Linear Regression Model
Interpreting Regression Output
•The F-ratio in the ANOVA table tests whether the
overall regression model is a good fit for the data.
The table shows that the independent variables
statistically significantly predict the dependent
variable, F = 944.111 Sig is < .05 (i.e., the regression model is a
good fit of the data).
Analysis of variance (ANOVA)
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
12. Multiple Linear Regression Model
Co-linearity diagnostics /Co-linearity issue
Interpreting Regression Output
• Colinearity (Multicolinearity) is the
undesirable situation where the
correlations among the independent
variable are strong.
•When two X variables are highly
correlated, they both convey
essentially the same
information. When this happens, the
X variables are colinear and the
results show multicolinearity.
•To check the issue Colinearity
diagnostics has been selected. The
Variation Inflation Factor and
Significant values determine the multicolinearity issue. If VIF value is greater than >2 or
>3 then the Multicolinearity exists and if Significant value >0.05 shows which
independent variable highly insignificant. The higher the significant value the higher the
insignificant independent variable. Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
13. Multiple Linear Regression Model
Interpreting Regression Output
•The higher the beta value shows the more influential the independent
variable and sign (Positive,Negative) shows the nature of the relationship
between independent and dependent variable. The Q5a has higher
influence.
•By having a glance on the sig column we can interpret. The significance
level should be below the <0.05.
•We can see that the all independent variables are <0.05 means that they
are statistically significance.
•The VIF of the all independent variables are <3 which shows that the
multicolinearity issue does not exist whereas the VIF should be near to
2 or <3.
Interpretation of Coefficients
Rizwan Manzoor (MSc Finance)
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16. 1. Increasing the sample size is a common first
step since when sample size is increased,
standard error decreases (all other things equal).
This partially offsets the problem that high
multicollinearity leads to high standard errors .
2. Remove the most intercorrelated variable(s)
from analysis. This method is misguided if the
variables were there due to the theory of the
model, which they should have been.
3. Take transformation of variables which is the
best fir for model.
Multiple Linear Regression Model
Interpreting Regression Output
Remedies of Multicolinearity
Rizwan Manzoor (MSc Finance)
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17. Multiple Linear Regression Model
Interpreting Regression Output
Remedy of Colinearity
1. Enter Method: It embraces all the variables that exist in the model and determines
whether the variables are significant or insignificant.
2. Stepwise Method: it embraces significant variables and excludes insignificant
variable.
3. Forward Method: It embraces the highly significant variables and order/rank
them but not to insignificant variables.
4. Backward Method: It will consider the highly insignificant variables and
order/rank them but not to significant variables.
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com
18. Using Enter method in Multiple Linear Regression, I
have selected a "best" model for predicting Satisfaction
level of staff of Educational Institution across Australia.
With this model, I found two Q4 and Q5 in the model
were high performing in all the across the institution,
while no predictors are insignificant.
The histogram showing the normality of data and PP
Plots also reflects the best fit of the model because all
the points are on the linear line
Multiple Linear Regression
Model
Summary:
Rizwan Manzoor (MSc Finance)
rizman2004@yahoo.com