Simple linear regression project

Vishrut Mehta
Japan Shah
SIMPLE LINEAR REGRESSION PROJECT (IE 5318)

Problem statement:
In this project of simple linear regression analysis, we are trying to determine the relationship between one response variable
(Y) and four predictor variables (X1, X2, X3, X4). We want to determine how the different values of all the predictor variables
affect the value of the response variable. And among all predictor variable, which variable shows the most linear relationship
with the response.
The variables:
The variables are as follows
 Response Variable (Y): Number of the followers of a person on the Twitter (in millions)
 Predictor Variable (X1): Number of tweets posted by a particular person
 Predictor Variable (X2): Number of years passed since that person has joined the twitter
 Predictor Variable (X3): Number of photos and videos posted
 Predictor Variable (X4): Number of people that person is following back
The data collection process:
We have used the official site of Forbes to find the first 100 most followed people on the twitter
(http://www.forbes.com/sites/maddieberg/2015/06/29/twitters-most-followed-celebrities-retweets-dont-always-mean-
dollars/#35671f137ef3)
Also we are using twitter to collect the data for the response variable and each predictor variables.
(www.twitter.com)
We searched for these people on the twitter for their accounts and verified them as their original account by looking at the
symbol , which is the symbol for the official account of persons on the twitter. We are using the first 40 most followed
people amongst them.
Why modeling this data set would be meaningful?
As we all know how much social media is affecting the lives of people in today's world and it is constantly evolving with
the new social media sites. It has revolutionized how we look at the things. While some people have made their way to
success through their hard work and years of experience, the others have made it through social media which also includes
the same amount of hard work. Amongst all of them today, the second most popular social media site is Twitter according
to ebizmba.com (http://www.ebizmba.com/articles/social-networking-websites). It is necessary to understand on which
factors the popularity of the people is driven. We are going to determine that among all these four predictor variables, which
variable affects the popularity of the person the most.

Scatter plots of the response variable vs each predictor variable:
Scatter plot of response variable Y vs X1
Here, there is a very less positive linear association between No of followers and No of tweets. There are no
outliers. There is no curvature. It is not strong predictor for the response variable. If we increase the no of tweets
by one then the no of followers would be increase very less.
Scatter plot of response variable Y vs X
Here, there is a positive linear association between No of followers and Years since they joined. There is no curvature and
outliers are not present. It is strong predictor for the response variable. We can say from the plot that years since they joined
impact the no of followers.
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No_of_followers
No_of_tweets
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No_of_followers
Years_Since_they_joined

Here, there is a positive linear association between No of followers and No of photos and videos posted. There is no
curvature. We have removed one outlier. After removing one outlier, the graph becomes positive otherwise there was a
negative linear association between response and explanatory variable.
Here, there is a positive linear association between No of followers and No of people following. There is no curvature and
there is one outlier.
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No_of_followers
No_of _photos and videos posted
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No_of_followers
No _of _people are following

Selecting one predictor variable:
From the scatter plot shown above, we can say that all the predictor variable are having positive trend. However, we
decided to take as “Years since they joined” as a predictor variable because it shows strong relationship with response as
compared to other predictors.

II.SIMPLE LINEAR REGRESSION MODEL
Data taken into the consideration
Y: No_of_followers
X: Years_Since_they_joined

The model form can be stated as follows:
Yi = β0 +β1Xi +εi
Where:
 Yi is the response variable
 β0 and β1 are parameters where β0 is the intercept and β1 is the slope
 Xi is predictor variable and is known constant
 εi is a random error with E (ε) = 0 and variance σ2
(εi)= σ2
, εi, εj are uncorrelated so that their covariance
is zero.
Fit a regression line

Software Output
Fitted model is (No_of_followers) Ŷ = -50.60528 + 12.52331X2 (Years_Since_they_joined)
The values for the parameter estimates are - β0 (Intercept): -50.60528, β1 (Slope): 12.52311
From the above output we can summarize as follows,
 Sum of squares indicates the amount of variability associated with each source. Model Sum of squares is 3195.
47982. This indicates the amount of variability explained by the model. Sum of square error (SSE) is 8254.40793.
This indicates an unexplained amount of variability. Corrected total (SSTO) is 11450. This indicates amount of
variability in the response.
 Mean square given as ratio of sum of squares and its corresponding degree of freedom. Mean square model is
3195. 47982. Mean square error is 217.22126, which is an estimate of the population variance.
 Root MSE is 14. 73843.This indicates that standard deviation of No of followers(Y) at each value of years since
they joined(X2)
 Departure Mean is 35. 49250. This indicates average of No of followers(Y) for the 40 observation.

 The coefficient of determination (R2
) is calculated as a ratio of sum of square model to the sum of square total.
Here R2
is 0. 2791. This shows that among all the variables, the variable X2(years since they joined twitter)
reduces the variability in our response variable (Y) by 27.91%.
 By taking the square root of R2
we get the value of coefficient of correlation (r) 0. 5282, which means that there is
52.82% linear relationship between our response variable and predictor variable.
 There is 41% variation in the sample mean compared to the mean of population.
 The standard error for the intercept and slope shows that how the variable would be if you resampled
 F value is 14.71, which is calculated by taking ratio of MSR and MSE. And the corresponding p-value is 0.0005.
Hypothesis:
Ho: β1 = 0 and H1:β1≠ 0
Decision rule:
If p < α then reject Ho
Conclusion:
From the table p= 0.0005 and α =0.05. As 0.0005 < 0.05, we reject Ho. It is a strong conclusion.
From this we can say that Years since they joined(x2) explains amount of variability in Y.

III. Inferences
A. Inferences on the parameter:
(Table 1.1)
From the above output we can say that we are 95% confident that the increment of the mean
number of followers lies between (5.913337, 19.13325) for each additional increase in the year.
B. Inferences on the True Line and Prediction:
Taking 95% confidence interval,
 CI (for mean response) = ŷ |x=7.5 ± t (1-α/2; n-2) S{ŷ h}
Now,
S {ŷ h} = sqrt [MSE * (1/n + (xn -–x̅ ) 2
/ ∑ ()2)
] = sqrt [217.22 * ((1/40) + (7.5-6.875)2
/ 20.375)] = 3.0975
ŷ |x=7.5 = -50.6052 + (12.5233 * 7.5) = 43.31955
CI = 43.31955 ± t (0.975, 38) *3.097
= 43.31955 ± (2.021 * 3.0975
CI = (37.659, 49.5797)

 PI (prediction interval) = ŷ |x=12 ± t (1-α/2; n-2) S{pred}
Now,
S {pred} = sort [ S2
{ŷ h} + MSE] = sqrt [9.5790 + 217.22] = 15.0598
PI = ŷ |x=7.5 ± t (0.975; 38) * 15.0598
= 43.31955 ± (2.021 * 15.0598)
PI = (37.059, 49.5795)
 The limits for Working-Hotellling confidence bands = ŷ |x=7.5 ± sqrt [ 2F (1-α, n-2)] *
S{ŷ h}
CB = 43.31955 ± sqrt [2F (0.95, 38)] * 3.0975
= 43.31955 ± sqrt [2* 3.15] * 3.0975
CB = (35.5479, 51.0911)
Confidence band limits at several xh along the range of x

Confidence bands with fitted line and data observations plot:
We are 95% confident that confidence band consists the entire fitted regression line.
IV. Model Assumptions
 The mean of the response variable is linearly related to the predictor variable.
 The probability distribution of the response variable (Y) doesn’t depend on the
level of explanatory variable X. So the variance of the response variable remains
constant.
 The error terms are normally distributed with mean 0.
 The error terms have equal variance.
 The error terms are uncorrelated.
 Conducting residual analysis to verify the model assumptions using plots. These
assumptions are:
I. The linear model is reasonable.
II. The residuals have constant variance.
III. The residuals are normally distributed.
IV. The residuals are uncorrelated.
V. No outliers.
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Y
X2
Y Vs X2

A. Box plot of residual
(Plot 1.1)
The box plot is basically used to find the symmetry of the residuals. Also it shows if
there are any outliers point in the residuals. From above graph, we can say that there
are two outliers.
B. Residual vs years since they joined twitter
(Plot 1.2)
By observing this graph, we conclude that there is no funnel shape and no curvature.
Which means that there is a constant variance in the residuals and there is linear
relation between our response variable and the predictor variable.
Outliers

C. Normal Probability plot of residuals
(Plot 1.3)
The normal probability plot is always made by plotting residual on Y-axis and their
expected values on the X-axis. Here our graph is right-skewed that is concave
upward shaped. This means the errors are not normal.
 Test for normality and constant variance
A. Test normality

H0: Normality is ok
H1: Normality is violated.
Decision rule: if ρ̂ < c (α, n) then reject H0.
Here c (α, n) = c (0.05, 38) = 0.972 which is less than ρ̂ (0.94567). Therefore, we
reject H0. Which is a strong conclusion. The normality is violated.
B. Modified Levene test for nonconstant variance:
In the modified levene test we divide the data into two groups. We have taken the
n1=35 for group1 and n2=5 for group2. The results from the SAS output of this test is
shown below.
Hypothesis:
H0 = variance is constant H1= variance is not constant
Decision rule:
If p<α then we reject H0. Here α = 0.05 and the value of p=0.4593. Which
means that p>α.
Conclusion:
Thus we fail to reject H0. Which means we accept H0. Thus the variance is
constant. It is a weak conclusion.

Overall discussion on which model assumptions are satisfied:
From the residual and x2 graph, we found that there is no curvature and there is no funnel shape.
So it proves that linear model is reasonable and there is a constant variance. Constant variance
assumption is also proved by the modified levene test.
Lack of fit test
Hypothesis
Ho: no lack in linear fit
H1: Lack in linear fit
Decision rule: p< α then reject Ho.
0.6979>0.05. So we fail to reject Ho and it is a weak conclusion .We can say that there is no lack
in linear fit.

Final Discussion
In simple linear regression analysis, we try to find the relationship between the two variables. One will be
the independent variable i.e. X and other will be dependent variable i.e. Y. Out of the four predictor
variable, we have select X2 i.e. Years Since they joined for the further analysis because it shows the
strong upward trend with the response. By performing statistical analysis, we came out with following
equation
(No_of_followers) Ŷ = -50.60528 + 12.52331X2 (Years_Since_they_joined). From various test, we can
say that there is linear relationship. Using residual plots such as residual vs x2, boxplot and normal
probability plot, we try to find what assumption satisfies. From graphs, we found that residuals are not
normal and there is a constant variance. From the lack of fit test, we can say that model is a good fit.
Ideas for future analysis
For the future analysis, we can take x4 (Number of people they are following back) as the predictor
variable because it has 47.85% correlation with our response variable. Although it is a good correlation,
the normal probability plot of this variable shows right-skewness, which means there may be chances of
having non-normality.
From all these points, this predictor variable may not prove to be an efficient variable to find the linear
relation with our response variable.
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Y
Sample Percentile
Normal Probability Plot

Simple linear regression project

Simple linear regression project

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