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0:02[Music]
0:07Hello, thanks for watching and welcome
0:10to the next video in my series on basic
0:12statistics. Now, as usual, a few things
0:15before we get started. Number one, if
0:17you're watching this video because you
0:19are struggling in a class right now, I
0:21want you to stay positive and keep your
0:23head up. If you're watching this, it
0:25means you've accomplished quite a bit
0:26already. You're very smart and talented,
0:29but you may have just hit a temporary
0:31rough patch. Now, I know with the right
0:33amount of hard work, practice, and
0:35patience, you can work through it. I
0:38have faith in you. Many other people
0:40around you have faith in you. So, so
0:43should you. Number two, please feel free
0:46to follow me here on YouTube, on
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0:52That way, when I upload a new video, you
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0:59that life is much too short and the
1:00world is much too large for us to miss
1:02the chance to connect when we can.
1:05Number three, if you like the video,
1:07please give it a thumbs up. Share it
1:10with classmates or colleagues or put it
1:12on a playlist. That does encourage me to
1:14keep making them for you. On the flip
1:17side, if you think there's something I
1:18can do better, please leave a
1:20constructive comment below the video,
1:21and I will take those ideas into account
1:24when I make new ones. And finally, just
1:27keep in mind that these videos are meant
1:29for individuals who are relatively new
1:31to stats. So, I'm just going over basic
1:34concepts, and I will be doing so in a
1:37slow, deliberate manner. Not only do I
1:40want you to know what is going on, but
1:43also why and how to apply it. So, all
1:46that being said, let's go ahead and get
1:49started.
1:51Hello and welcome to part three in our
1:53series on multiple regression. Now, I am
1:56going to assume you watched parts one
1:58and two of this series or you come to
2:00this video with that knowledge already
2:02in your head. So, if you need to go
2:05back, watch those two videos and then
2:07come back to this one. Otherwise, you
2:09may end up watching this one and become
2:11confused and frustrated. And neither of
2:13us want that to happen. So here in part
2:16three, we're going to talk about
2:17building regression models. So we will
2:20do that by introducing our independent
2:21variables into the model one at a time.
2:24Then we'll add them two at a time. And
2:26then in this case, we'll add all three.
2:28And then we'll see the models that come
2:30out of that process and then make a
2:32judgment call on which model is best. So
2:35we'll talk about all that criteria as
2:37far as determining the best model as we
2:39go forward. So all that being said,
2:41let's go ahead and get started.
2:46So as we discussed in parts one and two,
2:48conducting a multiple regression
2:50analysis requires a fair amount of
2:52pre-work before actually running the
2:54numbers in your software. So here are
2:56those steps real quickly. We generate a
2:59list of potential variables. So
3:01independence and the dependent variable.
3:03We collect data on those variables. We
3:06check the relationships between each
3:07independent variable and the dependent
3:09variable using scatter plots and
3:11correlations. So here we want to see
3:14which independent variables are actually
3:16related to the dependent variable in the
3:19first place because the ones that are
3:21not are probably going to be left out of
3:23the analysis.
3:25Then we'll check the relationships among
3:27the independent variables themselves
3:30using scatter plots and correlations. So
3:32here we're checking for multi-olinearity
3:36and then usually optional but I do it
3:38anyway and that is conduct simple linear
3:40regressions for each independent
3:42variable dependent variable pair and
3:45that's what we will do in this video
3:47because here we want to learn how to
3:48build models up and then tear them down
3:51to find the best one.
3:53Then we'll use the non-redundant
3:55independent variables in the analysis to
3:58find the best fitting model. And
4:00finally, most likely in a later video,
4:02we'll use the best fitting model to make
4:04predictions about the dependent
4:06variable. And we'll talk about
4:08confidence intervals and prediction
4:09intervals at that point.
4:14So remember, you are a small business
4:16owner that runs a delivery service like
4:18a courier service. You do same day
4:20deliveries of packages, letters, small
4:22cargo, etc. And you want to be able to
4:25predict the total travel time for each
4:29trip. So on a trip, you may have several
4:31deliveries. They may be close to your
4:33office, they may be far away, etc. So
4:36you go back and look at 10 past trips
4:38and record four pieces of information.
4:40The total miles traveled for that trip,
4:42the number of deliveries during that
4:44trip, the daily gas price, and finally
4:47the total travel time in hours, which is
4:50your dependent variable or the thing
4:51you're interested in predicting. So
4:54here's the data. We have miles traveled,
4:55that's our first independent variable,
4:57X1. number of deliveries during the
4:59trip, that's x2. And the gas price that
5:02day, that's x3. And then our dependent
5:05variable, travel time, is y there on the
5:07right.
5:11So remember, it's always a good idea to
5:13visualize the relationships. So we have
5:15our travel time dependent variable. Then
5:17we have three independent variables. So
5:19we have miles traveled, number of
5:22deliveries,
5:24and gas price. Now, of course, all three
5:26of those have some relationship to the
5:28dependent variable. We don't know that
5:29yet, but we will. But we also have to
5:31account for the relationships among the
5:33independent variables themselves, which
5:35are there in the dash line. So, we have
5:38six relationships we have to analyze and
5:40keep in mind as we do our analysis.
5:44Let's quickly review the scatter plots
5:46comparing the dependent variable and the
5:48independent variables individually. So
5:50the first scatter plot we have our
5:52travel time dependent variable versus
5:55our miles traveled or first independent
5:57variable and as you can see there we
5:59have a strong linear relationship starts
6:01at the lower left of the graph and goes
6:02to the upper right we have a correlation
6:04coefficient of 928 that's very very high
6:08we have a p value for the correlation of
6:090 which means it's less than 0.1 so we
6:14know that that variable miles traveled
6:17is related strongly related to our
6:19dependent variable able travel time. So
6:21we'll put a green check there.
6:24The second independent variable number
6:25of deliveries X2 that's also very
6:28strongly related. So the line starts in
6:30the lower left goes to the upper right.
6:32They fall along a rough line there. The
6:35correlation is 916. Again very high. The
6:38p value is less than 0001. So that is
6:41significant. So we'll put a green check
6:43there. So our first two independent
6:45variables do have strong linear
6:48relationships. very strong correlations
6:50with our dependent variable, which is
6:52good.
6:54Now, our third independent variable, gas
6:56price, does not. As you can see, the
6:59data points do not form any pattern.
7:01They're kind of all over the place. And
7:03we can see that in the correlation. It's
7:06267 with a p value of 0455.
7:10That is not significant. So, we'll put a
7:12red X there. So gas price X3 does not
7:17have any sort of linear relationship to
7:20the dependent variable right off the
7:22bat. So usually we would just go ahead
7:25and remove that from the model because
7:27if it doesn't have one to begin with,
7:29it's not going to contribute anything to
7:31the regression. But of course for now
7:34we're going to leave it in and see how
7:35it affects our numbers as we go forward.
7:40So now we have the scatter plots for the
7:42independent variable comparisons. So
7:44again, we're looking for
7:44multi-olinearity.
7:46So the first scatter plot, we have miles
7:48traveled versus number of deliveries.
7:51And here we have a problem. So our first
7:54two independent variables have a very
7:57very high correlation, a 0.956, a p
8:01value less than 0001. And I'll put a
8:04skull and crossbones there cuz this is a
8:06problem. two independent variables that
8:09are this highly correlated are going to
8:11cause some serious issues with our
8:13regression coefficients going forward.
8:16So, we'll keep that in mind as we go.
8:17I'll leave them in there for now so we
8:19can see how it affects things going
8:20forward.
8:22Now, miles traveled X1, gas price X3, no
8:25problems there. And then number of
8:28deliveries X2 and gas price X3, no
8:31problems there. So the only problem we
8:33have to worry about as far as
8:34multiolinearity goes is the correlation
8:37between the first two independent
8:38variables.
8:42So the quick summary. So the correlation
8:44analysis confirms the conclusions we
8:46reached by visual examination of the
8:48scatter plots. So we have some redundant
8:51multi-colinear variables. So miles
8:53traveled and number of deliveries are
8:55both highly correlated with each other
8:57and therefore are redundant. only one
9:00should be used in the regression
9:01analysis in the end. And we'll see how
9:04that all pans out as we build our model.
9:06We also have a non-contributing
9:08variable. So gas price is not correlated
9:11with the dependent variable at all and
9:14should probably be excluded right off
9:15the bat. Now again, for educational
9:19purposes, I'm going to leave all three
9:21variables in so we can see how putting
9:24them in there affects the regressions we
9:27do. But in the end, it'll all work out.
9:29You'll see.
9:32Okay. So, let's go ahead and get into
9:33some of our single variable regressions.
9:38Now, in this first step, we will perform
9:39a simple regression for each independent
9:42variable individually. The first will be
9:44conducted in Excel and then the rest in
9:46Mini Tab. That's what I prefer right
9:48now, but SPSS, SAS, Jump, R, etc. um or
9:52offline as well. You can get them all to
9:54generate basically the same output. So
9:57depending on which one you use, you
9:58should be fine. But I'll be focusing on
10:00a little bit of Excel and the rest
10:02MiniAB.
10:04We will discuss interpretations of the
10:06results we get from Mini Tab. And we
10:09will note how our results change. So
10:12we'll look at the coefficients in the
10:14regression. We'll look at the
10:16coefficient values. We'll look at their
10:19t statistics and we'll look at their p
10:21values.
10:22We'll look at the ANOVA table in the
10:24regression. So we'll look at the F value
10:27and the P value in that ANOVA table.
10:31We'll look and talk about the R 2, the R
10:342 adjusted and the R squar predicted and
10:37talk about what those mean. We'll also
10:39look at something called the VIF or the
10:41variance inflation factor and that's a
10:43statistic that MiniAB produces that will
10:45help us weed out multiolinearity.
10:49And we will also talk about something
10:51called malocp. That is a statistic that
10:54many tab outputs that will help us pick
10:56the best model in the end.
11:00So let's go ahead and look at our first
11:02regression. So we're going to regress
11:04travel time y which is our dependent
11:06variable on miles traveled which is our
11:08first independent variable. And here are
11:10our results from Excel. So we basically
11:12have three tables here. The first one in
11:14the top left are our regression
11:16statistics. Now in this case, multiple R
11:19is the same thing as our correlation.
11:21Since we only have one independent
11:23variable, they're the same thing. So
11:250.928, etc. is the same as the
11:28correlation we had a couple slides ago.
11:30Now R square is the proportion or
11:34percentage of variation in the dependent
11:37variable accounted for by the
11:39independent variable. So we can look at
11:42it as a percentage. So 86.15%
11:46of the variation in the dependent
11:48variable is accounted for by the
11:50independent variable. That's pretty
11:52high. Now the adjusted R square, that's
11:55the same thing as the R square.
11:57Obviously, it is just adjusted for the
11:59number of independent variables in our
12:01model, which in this case is one. So it
12:04will always be lower than the R squared.
12:07And how much lower really depends on the
12:09specific circumstances we're in. So the
12:12next number is the standard error of the
12:14regression. This is one of my favorite
12:16numbers, but unfortunately most people
12:19don't know how to use it or don't use it
12:21or just skip it or whatever else, but I
12:23think it's very helpful. So the standard
12:26error of the regression is the average
12:29distance of the data points from the
12:32regression line in dependent variable
12:35units. What we're saying here is that
12:37the data points are on average 342
12:43hours away from the regression line. It
12:46is in the units of the dependent
12:48variable and it gives us a measure of
12:51how tightly around the regression line
12:55our data points are. So it kind of forms
12:58a a channel or a band around the
13:03regression line. And the narrower that
13:05is, the more tightly our data points are
13:08around the regression. And the wider
13:10that band is, the more scattered they
13:13are from that regression line. So the
13:15standard error of the regression tells
13:17us relatively speaking how wide that
13:20band around the regression line is. And
13:23it's also helpful because it is in the
13:25units of the dependent variable. In this
13:27case, hours. And of course, we have 10
13:29observations. That's pretty
13:30self-explanatory. Now the ANOVA table
13:33that gives us the significance of the
13:35overall model. So we have an F statistic
13:38there of 49.768
13:40etc. with a p value of 0.00001
13:45that of course is significant. So the
13:48overall model here is significant. Now
13:51at the bottom we have some of our
13:53coefficient information. Now we're
13:55interested in the miles traveled
13:57coefficient. So under coefficients we
14:00have 0.0402
14:03that is the coefficient of our miles
14:05traveled and again that is in hours. So
14:08what we're saying there is that for
14:10every mile that's increased the time
14:14traveled increases by 042
14:18hours. Then we have the p value which is
14:210.1. So we know it is significant. Now,
14:24if you notice in this case, the p value
14:26for the miles traveled coefficient is
14:29the same as the p value for the ANOVA.
14:32And that's because we only have one
14:34independent variable. So, how can we use
14:36this information? Well, we can take our
14:39coefficient information at the bottom
14:40and generate our regression equation.
14:43So, we have an intercept of 3.1855, and
14:45again, that's rounded up here at the
14:47top. Plus 00403.
14:50That is our coefficient for miles
14:51traveled down here at the bottom. and
14:53then times the miles traveled. That's
14:55our independent variable. That's our x1.
14:58So we have 3.1856
15:00plus 0403
15:03x1 where x1 is miles traveled. So what
15:06does that mean? An increase in 1 mile.
15:10So miles traveled x1. So one mile will
15:13increase delivery time by 043
15:18hours. And that's how we can interpret
15:20the coefficient in this simple
15:22regression. Now, let's go ahead and make
15:24a rough prediction for miles traveled
15:27that's sort of within the range of our
15:29original data. So, we'll pick an 84 mile
15:32trip estimate. So, we go ahead and
15:35substitute 84 in for our x1. That gives
15:38us 6.5708
15:41hours. So, that's a very rough estimate
15:44of how long it would take for an 84 mile
15:47trip. So 6 hours and 34 minutes.
15:51But remember this is just an estimate.
15:53So it's going to have an interval around
15:55it. It's going to have some error around
15:57it. Now we can go ahead and find that
15:59prediction interval using some things we
16:02already know. And again what I've done
16:04here is a very rough sort of estimate.
16:07Mini tab can give us exact numbers but a
16:10rough estimate of the prediction
16:11interval for an 84 mile trip. So we have
16:146.5708
16:16plus or minus 2.31.
16:19Well, where do I get 2.31?
16:22That comes from our t distribution, our
16:24t table. So in this example, we have n
16:28minus 2 degrees of freedom. So that's 10
16:31minus 2 in this case because we have 10
16:33observations. So we go to our t table.
16:36We look at degrees of freedom of eight.
16:39Then we look down the table for a 95%
16:42interval and we have a critical t of
16:452.31. That's where that comes from. And
16:48then we have 3423.
16:51Where does that come from? Well, that's
16:54my magic number over here on the left.
16:56The standard error of the regression. So
16:58this is very much like any other
17:00interval we calculated back in interval
17:03estimation. So we have a point
17:05estimator. So 6.5708
17:08plus or minus the t alpha / 2 which is
17:122.31
17:14times the error which in this case
17:16is.3423.
17:19I don't expect you to sort of get this
17:20right now but I just want to show you
17:21sort of how we use what we have here. So
17:24that creates an interval of 5.7764
17:28to 7.3615
17:30hours or 5 hours 47 minutes to 7 hours
17:36and 22 minutes. That's our 95%
17:39prediction interval for an 84 mile trip.
17:43So again, that's sort of a step forward
17:46what we're going to do in future videos,
17:48but I just wanted to quickly show you
17:50how we can use the information we get
17:53from a regression to make some rough
17:55predictions for other values.
17:59So here is our second one toone
18:01regression. So we have our second
18:03independent variable number of
18:05deliveries and our dependent variable
18:07travel time. And this comes from Mini
18:09Tab. Now obviously it looks a little bit
18:11different. Now, I will say that one of
18:14the best skills to have if you're doing
18:16statistics or whatever else is to be
18:18able to use really any software package
18:21or at least look at the results of any
18:24software package and know how they
18:27correspond to each other. So, in this
18:29case, the F value of 41.96
18:34that's along the regression line is the
18:38same F value we had in Excel in the
18:40previous slide. So we look at the
18:42regression line going across. We have an
18:45F value of 41.96
18:48and a P value of 0. So we know that's
18:51significant. Now at the bottom we have
18:54the model summary. So S is the same
18:57thing as the standard error of the
18:59regression that we had in Excel. So here
19:02it's 368091.
19:04Then again we have R 2 we have R 2
19:08adjusted
19:09and then we have in this case R 2
19:13predicted. So this is an addition that
19:16MiniAB gives us and R 2 predicted is
19:21basically how well our model does at
19:26predicting
19:27additional data points. So we'll talk
19:29about that more as we go. But R squ is
19:32really about predictive power.
19:35So here are our coefficients. So again,
19:37this looks very similar to what we had
19:39in Excel. So we'll look at the number of
19:41deliveries uh row. So we have a
19:44coefficient of 4983. We have a t value
19:47of 6.48. Its p value is 0. So we know it
19:52is also significant. The vif we'll talk
19:55about later. And then the regression
19:57equation, which is really nice about
19:59many tab and other software packages, it
20:01actually gives us the regression
20:02equation. So we have 4.845.
20:05So that comes from the constant
20:07coefficient up there at the top.
20:09Plus4983
20:11that comes from the number of deliveries
20:13coefficient up there. And then we
20:15multiply that by the number of
20:16deliveries we actually have. So an
20:19increase in one delivery, one additional
20:22number of deliveries will increase
20:24delivery time by 4983 hours or almost a
20:30half an hour. Again, that's how we
20:32interpret a simple regression
20:34coefficient.
20:37So, let's go ahead and make a rough for
20:39delivery estimate. Now, we're not going
20:40to do the prediction interval again.
20:42We'll just do the four delivery
20:43estimate. So, let's say a trip has four
20:45deliveries. So, we can go ahead and
20:47substitute the four into our regression
20:49equation and we come up with an estimate
20:52of 6.838
20:54hours or 6 hours and 50 minutes. So if
20:59we come into work one day and we have a
21:01trip with four deliveries on it. So
21:04based on our data, the best estimate we
21:06have is a trip of 6 hours and 50
21:09minutes.
21:12So finally, here's our last single
21:14regression. So gas price X3 and travel
21:17time Y. So we go across the regression
21:20line here. We have an F value of
21:2462.
21:26That's very low. Then we have a p value
21:29of 0.455.
21:31That is not significant. That's
21:33obviously not below 0.05. So that
21:36basically confirms what we looked at in
21:38the scatter plots and correlations. And
21:40that is that gas price really has no
21:43relationship or no linear relationship
21:45to travel time. Now if we look down at
21:48the model summary, we have a standard
21:50error of regression of86
21:53hours. That is a huge standard error of
21:56the regression. We have an R squar
21:58that's 7.14%.
22:00You know why even bother? The R squ
22:02adjusted is literally zero and the R
22:04square predicted is literally zero. So
22:07we know that this variable is really a
22:10dud.
22:11So here are our coefficients. We have
22:14the gas price coefficient of 81. We have
22:18a T value of 78 very low obviously P
22:22value of 0455. Same thing as we have in
22:24the f value above the p value up there.
22:27We know that's not significant. The
22:30regression equation, why even bother?
22:32This is really not worth exploring any
22:34further. We know that gas price is
22:36really not a variable that contributes
22:38to travel time.
22:41Let's go ahead and summarize these three
22:43models. So in the first line, we have
22:45the case where we're only using our
22:47first independent variable. We had an F
22:50of 49.77,
22:52a p value that's very low, significant,
22:54less than 0001. We had a standard error
22:57of the regression of 34230.
23:01We had an R squar of 84.42
23:04and an R squar predicted of 79.07.
23:08Now I did that in many tabs, so we had
23:10it um but that's the actual R square
23:13predicted that was produced. And then in
23:16the second case where we had the second
23:18independent variable we had an f of
23:2041.96
23:22p value less than 0001.
23:25Now here we had a standard error of the
23:27regression of 36809.
23:31So remember what I said. What does that
23:33tell you? Well, in the first case in the
23:36um the top row, on average, our data
23:39points were 34230
23:42hours away from the regression line. In
23:46the second case, they were 36809 hours
23:50away from the regression line. So, which
23:53model has data points that are more
23:55clustered in around the regression line?
24:00Well, the first one does because it
24:01standard error of the regression is
24:03lower. Now, the R squ adjusted for the
24:05second one is 81.99.
24:08It's a little bit lower than the one at
24:09the top. And then the R square predicted
24:12is 70.27, which is quite a bit lower
24:15than the one at the top. And then
24:17finally, we sort of have our our dud,
24:20our no good one, where we're using X3.
24:22I'm not even going to go through that
24:24anymore because it's obviously um no
24:26good. So, if we look at this example
24:28where we're only using one independent
24:31variable, which one do you think is the
24:34best?
24:35Well, I would have to say that the first
24:36one's the best. We have the highest
24:39F-stistic. We have the lowest standard
24:42error of the regression. We have the
24:44highest R 2 adjusted. And we definitely
24:46have the highest R square predicted. So
24:49of these three, the first one is
24:52definitely I would say the best.
24:57So let's do the case where we have two
24:59variables. So now we're going to put two
25:01variables in at a time. So we'll put X1
25:04and X2 into the regression. We'll put X1
25:06and X3 into the regression. And we'll
25:08put X2 and X3 into the regression. So
25:11this is a subset of two variables but
25:13the interpretation is basically the

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Use the transcript to take notes, quote the speaker, build a study guide, generate a summary with ChatGPT or Claude via the YouTube Summary tool, or export it as a timed subtitle file with YouTube to SRT. You can also re-open it in the transcriber to translate the transcript into 100+ languages.

Free YouTube transcript tool

YouTube2Text is a free YouTube transcript generator — no signup, no daily limit. Paste any YouTube link and get the full transcript instantly, with timestamps, click-to-jump, translation to 100+ languages, AI prompts for ChatGPT, Claude, and Gemini, and exports to TXT, SRT, VTT, or Markdown.