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<h1 class="title toc-ignore">Mutiple Linear Regression</h1>
<h4 class="author">Yumeng Qi & Jiying Wang</h4>
</div>
<p>In this section, our objective is to elucidate the connection between
flight-related information and flight delays within the framework of
linear regression. Adhering to the model’s assumptions, our initial step
involves computing pairwise correlation coefficients to mitigate the
impact of multicollinearity. Pairs exhibiting coefficients exceeding 0.8
are deemed highly correlated, signifying the potential to introduce
distortion into the model. Subsequently, we conducted stepwise
regression selection, minimizing the Akaike Information Criterion (AIC).
This stepwise approach assists in refining the model by iteratively
adding or removing variables, ensuring a more parsimonious and
statistically robust representation of the relationship between the
variables and flight delays.</p>
<div id="correlation-coefficient" class="section level1">
<h1>Correlation Coefficient</h1>
<p>Upon examining the plot, we observed two pairs with correlation
coefficients surpassing 0.8: namely, wind_gust and wind_speed, and
air_time and distance. The presence of highly correlated pairs suggests
a potential for one variable to be essentially substituted by another in
the model, which leads to the problem of multicollinearity. In the
presence of multicollinearity, the estimates of individual regression
coefficients become sensitive to small changes in the data. This
sensitivity can lead to coefficients with large standard errors and
reduced statistical significance. Also, it inflates the variance of the
regression coefficients, making them less precise. This increased
variance can result in wider confidence intervals, making it more
challenging to draw reliable conclusions about the effects of individual
predictors.</p>
<pre class="r"><code>df_2013_mlr = df_2013_mlr |>
mutate_if(is.numeric, scale)
M = cor(df_2013_mlr |> select_if(is.numeric))
corrplot(M, order = 'AOE', type = 'upper', tl.pos = 'n', method = "number")
corrplot(M, add = TRUE, type = 'lower', method = 'ellipse', order = 'AOE',
diag = FALSE, tl.pos = 'l', cl.pos = 'n',tl.col='black',tl.cex = 0.8)</code></pre>
<p><img src="MLR_files/figure-html/scale-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>#corrplot(cor(df_2013_mlr |> select_if(is.numeric)), method = "number")
# delete distance and wind_speed for less unqiue value</code></pre>
<p>In response to this correlation finding, we conducted an assessment
based on the unique values of each variable within these pairs. The
rationale was to retain the variable with a greater number of unique
values, indicative of a broader range of information. Consequently, we
decided to eliminate wind_speed and distance from the analysis, aiming
to enhance the model’s stability and interpretability by resolving
multicollinearity issues.</p>
<table>
<colgroup>
<col width="6%" />
<col width="23%" />
<col width="26%" />
<col width="21%" />
<col width="21%" />
</colgroup>
<thead>
<tr class="header">
<th>highly correlated variables</th>
<th>wind_gust</th>
<th>wind_speed</th>
<th>air_time</th>
<th>distance</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>unique cnt</td>
<td>33</td>
<td>30</td>
<td>462</td>
<td>177</td>
</tr>
</tbody>
</table>
</div>
<div id="stepwise-regression" class="section level1">
<h1>Stepwise Regression</h1>
<p>In the context of the linear model, we have taken several important
preprocessing steps, including standardizing the variables, employing
one-hot encoding to handle categorical variables, and excluding highly
correlated pairs. These measures were implemented to enhance the
stability and interpretability of the model.</p>
<p>With a heightened level of confidence in the prepared dataset, we
proceeded to conduct stepwise regression. This method systematically
refines the model by iteratively selecting or removing variables based
on criteria such as the Akaike Information Criterion (AIC). The goal was
to precisely capture the relationships between flight delay time and
various flight information parameters.</p>
<p>By leveraging stepwise regression, we aimed to ensure that the final
model includes the most relevant and significant predictors, thereby
providing a streamlined and statistically robust representation of the
intricate relationships within the data.</p>
<div id="variable-estimation" class="section level2">
<h2>Variable Estimation</h2>
<p>The plot below unveils insights into the top 20 estimated values of
the predictors. Strikingly, these top 20 predictors exclusively consist
of destination dummy variables. Notably, only Daniel K. Inouye
International Airport (HNL) stands out with negative estimates, implying
that flights to HNL are less likely to experience delays—a delightful
discovery. Perhaps both visitors and flight crews couldn’t wait to join
the beach!</p>
<pre class="r"><code>fit_AIC = lm(formula = arr_delay ~ origin + hour + carrier + dest + air_time + minute +
wind_dir + pressure + visib + arrival_date,
data = df_2013_mlr)
#summary(fit_AIC)
#stargazer(fit_AIC, title="Regression Results",type='html', single.row = FALSE, ci=TRUE)
coeff <- tidy(fit_AIC) %>%
arrange(desc(abs(estimate))) %>%
filter(term != '(Intercept)') %>%
slice(1:20)
ggplot(coeff,
aes(x = reorder(term,estimate), y = estimate)) +
geom_col(fill = 'skyblue') + coord_flip()+
geom_text(aes(label = round(estimate,2)))+
xlab('Predictors')+
ylab('Estimate')+
theme(legend.position = "none") + theme_minimal()</code></pre>
<p><img src="MLR_files/figure-html/estimate%20plot-1.png" width="672" style="display: block; margin: auto;" />
According to a <a
href="chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.transportation.gov/sites/dot.gov/files/2023-04/February%202023%20ATCR_Revised.pdf">Air
Travel Consumer Report</a> relesd by the U.S. Department of
Transportation in Dec, 2022 (due to limited resources, we failed to
trace the rank from 2013), the top 10 U.S. Airports with the Most Delays
are:</p>
<ol style="list-style-type: decimal">
<li>Midway Airport, Chicago (MDW, 47% delayed)</li>
<li>Stapleton Airport, Denver (DEN, 43%)</li>
<li>Baltimore-Washington International (BWI, 42%)</li>
<li>Love Field, Dallas (DAL, 42%)</li>
<li>Seattle-Tacoma Airport (SEA, 38%)</li>
<li>Harry Reid International, Las Vegas (LAS, 38%)</li>
<li>Orlando International (MCO, 37%)</li>
<li>Fort Lauderdale Airport (FLL, 37%)</li>
<li>Newark Liberty International (EWR, 35%)</li>
<li>Nashville International Airport (BNA, 32.9%)</li>
</ol>
<p>So, the funky thing is, the cool airports that made it to the top 20
in 2013 don’t exactly jive with the list of the top 10 delay-prone
airports. It’s like they’re dancing to different tunes! Why, you ask?
Well, one possibility is that our 2013 dataset is a bit of a New York
City party, and you know how NYC likes to do things its own way. It’s
like the dataset is saying, “I’m from NYC, and I do delays
differently!”</p>
<p>Or maybe, some airports might have gotten their act together in the
past 10 years.</p>
<p>So, in the grand scheme of things, our dataset from 2013 might be a
bit like a vintage record – charmingly outdated, but still groovy. And
those airports with the top delays? Well, they could be the cool kids
who’ve learned a thing or two about punctuality since then. Life’s a
journey, and apparently, so are our flights!</p>
</div>
<div id="model-summary" class="section level2">
<h2>Model Summary</h2>
<p>The extensive table presented below provides estimates and confidence
intervals (CIs) for each variable incorporated into the stepwise model.
Although the table may initially appear lengthy and somewhat lacking in
informativeness, a closer examination reveals an adjusted <span
class="math inline">\(R^2\)</span> value of only 0.110. This modest
value suggests that the linear model employed may not be sufficiently
adept at elucidating the potential for flight delays.</p>
<style>
table {
margin-left: auto;
margin-right: auto;
}
caption {
caption-side: top;
text-align: center;
font-weight: bold;
}
</style>
<table style="text-align:center">
<caption>
<strong>Regression Results</strong>
</caption>
<tr>
<td colspan="2" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
<em>Dependent variable:</em>
</td>
</tr>
<tr>
<td>
</td>
<td colspan="1" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
arr_delay
</td>
</tr>
<tr>
<td colspan="2" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
originJFK
</td>
<td>
-0.108<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.138, -0.079)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
originLGA
</td>
<td>
-0.065<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.089, -0.040)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
hourEvening
</td>
<td>
0.268<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.247, 0.288)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
hourMorning
</td>
<td>
-0.173<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.193, -0.154)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
hourNight
</td>
<td>
0.181<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.083, 0.278)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierAS
</td>
<td>
-0.278<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.467, -0.090)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierB6
</td>
<td>
0.162<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.122, 0.201)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierDL
</td>
<td>
-0.048<sup>**</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.086, -0.010)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierEV
</td>
<td>
0.220<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.170, 0.271)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierF9
</td>
<td>
0.282<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.135, 0.429)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierHA
</td>
<td>
-0.343<sup>**</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.624, -0.061)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierOO
</td>
<td>
-0.074
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.904, 0.755)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierUA
</td>
<td>
0.003
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.035, 0.040)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierVX
</td>
<td>
-0.026
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(-0.096, 0.044)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
carrierWN
</td>
<td>
0.077<sup>**</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.014, 0.140)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
destACK
</td>
<td>
3.683<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(3.140, 4.225)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
destALB
</td>
<td>
3.929<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(3.468, 4.390)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
destATL
</td>
<td>
2.670<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(2.261, 3.078)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
destAUS
</td>
<td>
0.743<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.339, 1.148)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>