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0:00So let's do the case where we have two
0:02variables. So now we're going to put two
0:04variables in at a time. So we'll put x1
0:07and x2 into the regression. We'll put x1
0:09and x3 into the regression. And we'll
0:11put x2 and x3 into the regression. So
0:14this is a subset of two variables, but
0:16the interpretation is basically the
0:20same. So this first one we have travel
0:23time, our dependent variable versus
0:25miles traveled together with number of
0:28deliveries. So our x1 and
0:31x2. Let's look at our regression here.
0:34We have an f value of
0:3823.72 with a p value of
0:410001. Now for now I want you to ignore
0:44the f value and p value for the
0:48individual variables. We'll get to that
0:50in a later video. So just look at the
0:52regression line across the top here. So
0:5523.72 F value P value of 0001. Of course
0:59that is significant. Now let's look down
1:02below. We have a standard error of the
1:05regression of
1:08352642. So that's again that's in the
1:10ballpark of what we had in the one
1:13variable models. We have an R squar of
1:1687.14, an adjusted R squar of 83.47%
1:2047% and then an R squar predicted of
1:2659.95%. So that's significantly lower
1:28and we'll talk about what that means
1:30later in the video. So now let's look at
1:33the coefficients and this is where
1:35things get interesting. So let's look at
1:38the miles traveled coefficient. So its
1:42value is
1:430262. Its t value is
1:471.31 and its p value is
1:52232. That is not significant. It's not
1:56below 05. Now look at number of
1:59deliveries. Coefficient
2:01of84, t value of 73 and a p value of
2:08487. That is not significant either.
2:12So, here's the weird thing. We have an
2:15overall model that is significant. Okay,
2:19up the top we have an F value of
2:2223.72 and we have a p value of
2:250.001. But down below, neither of our
2:29coefficients are statistically
2:31significant. Very strange. And then of
2:34course down below we have our regression
2:37equation. So remember what we discussed
2:39before, these two variables X1 and X2
2:43are extremely correlated with each
2:46other. So they they're almost on a
2:48straight line that goes from bottom left
2:50to top right. The correlation is N56.
2:54That is very significant. So these two
2:57variables are
2:59multicolinear. Now look what happens. We
3:02have two variables that are collinear.
3:05The overall model is significant, but
3:08the individual coefficients are not. Now
3:11you see what happens when we put two
3:14variables in a regression that are
3:17correlated, that have high levels of
3:20colinearity. The coefficients in the
3:23regression model go haywire. They go
3:26crazy. So this is why we look to
3:27eliminate variables from the get-go.
3:30Now, Mini Tab also has this variance
3:33inflation factor or the VIF. Now, how
3:36this helps us is that it points out
3:39variables that are collinear that points
3:43out
3:44multi-olinearity. So, we'll talk about
3:45it more as we go, but a VIF of
3:5011.59 should send alarm bells off um in
3:53our statistical minds. We should know
3:56that is a serious problem. And this
3:59model we have in front of us is very
4:01suspect. And we can base that on several
4:04criteria. We know from our scatter plots
4:07that they're highly correlated, very
4:09highly correlated. We have this weird
4:11situation where the overall model is
4:13significant, but the individual
4:15coefficients are not. And then we have
4:18this VIF that's through the roof. So all
4:21that taken together should tell us that
4:24this model has some problems. Now, we'll
4:26keep it in here just for learning
4:28purposes, but just keep in mind the
4:30criteria here. And I also want to point
4:33out that when we have an R squar
4:36adjusted, that's
4:3983.47%. Then the R squar predicted falls
4:42off a cliff. Now it's at
4:4559.95%. That tells us that we have a
4:49serious problem in our model. So all
4:52that taken together lets us know that
4:53something is wrong.
5:00So in our ANOVA table we have a
5:02regression line here. We have an F value
5:04of
5:0522.63. That is a p value of 0.001. So
5:09that is significant. Down here in the
5:11model summary, we have a standard error
5:13of the regression
5:15of.359883 hours. Of course, we have an R
5:18squared of 86.61% 61% R squared adjusted
5:22of
5:2382.78% and an R squar predicted of
5:2868.11%. Now notice couple things here.
5:31Of course, our ANOVA table overall is
5:34significant. Our R squar is adjusted as
5:37very high and our R 2 predicted doesn't
5:41fall off a cliff like we saw in the
5:44previous example with X1 and X2.
5:47So here are our coefficients. So in this
5:50case we have miles traveled which is
5:530.04137. That's our coefficient for
5:55miles traveled. T value of 6.44 with a p
5:58value 0. So that's fine. Now look at gas
6:03price. We have a negative coefficient
6:08-2.19. Now think about what this means.
6:11So what this is saying is that if we
6:14hold miles traveled constant, that's our
6:17x1. If we hold that constant, and we
6:20increase the price of gas a dollar, then
6:25the travel time will decrease by 219
6:29hours. So gas price goes up and the
6:33travel time goes down. Does this make
6:36any real sense to me? It does not. So, I
6:40don't know about you, when gas prices go
6:41up, I drive slower. But this is saying
6:44the travel time goes down. And what this
6:48points out is that we are putting this
6:50gas price variable in the regression,
6:52but it doesn't have any real
6:54relationship to the dependent variable.
6:57Remember, so now we get some very weird
7:00coefficients down here at the bottom. So
7:02that's why we try to eliminate variables
7:04up front because it really messes up the
7:07coefficients that come out of the
7:08regression process. So let's interpret
7:10both of these coefficients. First, miles
7:12traveled. So if gas price is held
7:15constant, then travel time is expected
7:17to increase by
7:2104137 hours for each additional mile
7:24traveled. Now does that make sense in
7:27real life? Well, yes it does. So if I
7:29travel further more miles, I expect my
7:32travel time to go up. Now how about gas
7:35price? If miles traveled is held
7:37constant, then travel time is expected
7:40to decrease by 219 hours for each
7:45additional dollar increase in gas price.
7:48And again, that really does not make any
7:50sense in real life and in statistics
7:54either because this coefficient is just
7:56kind of weird. So again, we're going to
7:58keep this one kind of off to the side,
8:00noting we have some weird coefficients
8:02in the gas price, and that's probably
8:05because we included a variable in the
8:08regression that has no relationship to
8:10the dependent variable to begin with. So
8:12let's go ahead and do the last two
8:15pair. So this is X2 and X3. So numbum
8:18deliveries and gas price. So look at the
8:21ANOVA table, the regression line. We
8:23have an F value of 27.63 63. P value is
8:280. So we know that's significant. We
8:31have a standard error of regression down
8:33here at the bottom of
8:36329703 hours. R squared of
8:4088.76%, R squared adjusted of
8:4385.55%. And an R squar predicted of
8:4871.76%. So those are all pretty high.
8:50The R squared predicted does not go off
8:52a cliff like we saw in the first
8:54example. So, so far things don't seem
8:56too crazy. Now, look at our
8:59coefficients. Uh-oh, we have the same
9:02problem again. So, number of deliveries,
9:04that looks fine.
9:05So,.5665, T value of 7.13, P value 0.
9:11Fine, looks good. Gas price has gone
9:13negative again.
9:16So,.765, T value
9:19of.172. The P value is not significant
9:22in this case. So we have this weird
9:25situation again where the gas price
9:28coefficient went
9:30negative. So interpret these again. If
9:33gas prices is held constant then travel
9:35time is expected to increase
9:38by.5665 hours for each additional
9:41delivery. Now does that make sense in
9:43real life? Well yes. I expect the travel
9:47time to go up for each additional
9:49delivery I have to make. Now how about
9:52the gas price problem? If number of
9:55deliveries is held constant, then travel
9:57time is expected to decrease by 765
10:01hours for each additional dollar
10:03increase in gas price. That doesn't make
10:06any sense. So again, we have this
10:08problem where we included a variable in
10:11the model that has no natural relation
10:14to the dependent variable whatsoever. It
10:16messes up our coefficients and really
10:18this model is no good. So let's go ahead
10:21and summarize these three
10:25models. So the top three lines are the
10:28first three models we did. So that's our
10:30single variable models. Now let's look
10:33at the second three. So in our first two
10:36variable model, we had x1 and
10:38x2. So we had an f of 23.72.
10:43Now we expect the fs to be about the
10:46same for each one variable model and
10:49each two variable model etc. Okay. So
10:5223.72 we have a p value 01. That's fine.
10:56Now we have a standard error of
10:58regression of
11:0235264. So remember what that tells us
11:04that tells us how tied in our data
11:09points are to the regression line. So in
11:11this case they are on
11:14average.35264 hours away from the
11:17regression line and you can compare that
11:18to the ones we have above. So our R
11:22squar is
11:2383.47. Our R square predicted is
11:2659.95. That's a huge drop off from the R
11:29squ adjusted. Then we have this VIF over
11:33here of
11:3511.59. That is huge and that is a
11:37problem. And that's because x1 and x2
11:41are collinear. That's the problem we
11:43have there. Now the second one from the
11:45bottom that's x1 and x3. So we have
11:4822.63 for the f0001 for the p value. The
11:52standard error of the regression 35988
11:56hours. And then we have R squ adjusted
11:58at
11:5982.78%. R square predicted
12:0268.11. Everything looks pretty much okay
12:05there. Then we have a VIF of 1.14.
12:08That's not a problem. But remember from
12:10our
12:11coefficients, we had a negative X3
12:14coefficient, which is very weird. So
12:17even though everything in that row looks
12:20okay, we know that we have a coefficient
12:22oddity. So we have to keep that in mind.
12:24And then finally here we have the X2X3
12:28model. So 27.63.
12:31Then we have the p value less than 0001.
12:34Standard error of the regression of
12:3732970. Again, that's in the ballpark of
12:39everything else. But if you look above
12:42it, you can see that so
12:44far that is the best fit around the
12:48regression line. So on average
12:5232970 hours away from the regression
12:55line, R squared adjusted of 85.55%.
13:00Now look at that column. That's the
13:03highest adjusted R squar we've had. Now
13:06go over to the R square predicted.
13:08That's 71.76. There's nothing really
13:10spectacular there relative to everything
13:12else. And then of course a VIF of 1.33.
13:16No problem there. So we have to decide
13:19here. We have this last one with a
13:23higher F than the two above it. We have
13:26a smaller standard error of the
13:28regression, which is what we'd like to
13:30see. We have a relatively high R squar
13:32adjusted at
13:3485.55%. In fact, it's the highest in
13:36that column there. And the R square
13:38predicted is what we'd expect. But
13:42remember from the coefficients, this is
13:45another example of where we have a
13:48negative coefficient. We have a negative
13:52gas price coefficient. So even though
13:54everything looks okay here, we also have
13:57to keep in mind our coefficients from
13:59the previous step. So that might be a
14:01problem. So let's go ahead and define
14:03what VIF actually is. And I just quoted
14:07this from many tabs blog. The URL is
14:10down here at the bottom. Now let's go
14:11ahead and quickly read what it says. So
14:14one way to measure multiolinearity is
14:16the variance inflation factor or the VIF
14:20which assesses how much the variance of
14:23an estimated regression coefficient
14:25increases if your predictors your
14:28independent variables are
14:30correlated. If no factors are correlated
14:33if no independent variables are
14:35correlated the VIFs will all be one. Now
14:40let me pause there. Look at the VIFs for
14:43the first three models. They're all
14:44exactly one. Well, why is that? Well,
14:48there's only one independent variable in
14:49them. So, they're going to be one. There
14:52is no correlation there. So, they'll all
14:54be one. Now, a VIF between five and 10
14:58indicates high correlation. That may be
15:01problematic. So, do we have any between
15:03five and 10? Uh, nope. Not so far. Now
15:07if the VIF goes above 10, you can assume
15:11that the regression coefficients are
15:13poorly estimated due to
15:17multicolinearity. So look at the first
15:19two variable model. We have a VIF of
15:2311.59. Now remember why that is. Our two
15:27independent variables X1 and X2 had a
15:31correlation above N5. They were
15:34extremely highly correlated. So that VIF
15:38the variance inflation factor points out
15:40that hey you have a problem there you
15:43have some multiolinearity some severe
15:46multiolinearity in that model and
15:48therefore we would just ax that model
15:51out of the we would just forget it so
15:53we'll leave it there for now but just
15:55know that the vif helps us find
16:00multiolinearity okay and finally the
16:03full model we're going to throw in all
16:05three independent depent variables and
16:07see what
16:09happens. Okay, so here is the ANOVA
16:12table from Mini Tab for all three
16:14independent variables. So let's look at
16:16the regression lineup here. We have an F
16:19value of
16:2116.99 with a p value of 02. So the
16:26overall model is significant. So the
16:28model summary, we have a standard error
16:30of the regression of
16:33344694 hours.
16:35We have an R squared of
16:3789.47, an R squared adjusted of
16:4184.2% and an R 2 predicted of
16:4657.49. Now, what's the red flag there?
16:49The R squared adjusted was
16:5184.20. The R square predicted is
16:5757.49. That's a huge drop
17:01off. So, the coefficients real quickly.
17:04So miles traveled had a coefficient of
17:080141. Numb deliveries was 383. Then we
17:11have the strange gas price
17:14that's.607 again. Now if we look at our
17:17p values, it gets even more strange. So
17:20the p value for miles traveled is 548.
17:24That is not significant. The p value for
17:26numbum deliveries
17:27is.249. Not significant. Guest price
17:31293. Not significant. even though it
17:34really doesn't matter because that's a
17:36junk variable at this point. Now, if you
17:38look at the VIFs, look at
17:42those. For miles traveled, it's
17:4614.94. For number deliveries, it's
17:5017.35. So, what does that tell us? We
17:53have severe severe problems with
17:57multiolinearity in this model. Severe
18:00problems. basically terminal death
18:03problems. But we go ahead and have the
18:05regression equation down here at the
18:06bottom just for kicks I guess. But
18:09basically this model is
18:14junk. So here are all of our models put
18:17together. So we're getting to the grand
18:19finale finally. So at the bottom we have
18:22this new model with an f of 16.99 p
18:25value
18:2602. Uh the standard error of the
18:28regression
18:3134469 R squared adjusted
18:3484.2%. R square predicted
18:3857.49%. Then we have our VIFs. I put
18:42those below each variable because I ran
18:44out of room. So for X1 it was 14.94, X2
18:4817.35, X3
18:511.71. So we can see that that's a
18:54problem. So step back and look at this
18:56last one again. We can see that we have
18:57a huge drop off from the R squared
19:00adjusted to the R square predicted just
19:02like we do at the top of the two
19:04variable models where we went from 8347
19:07to
19:0859.95. Now here is the question. Which
19:12model is the
19:14best? So let's start with knocking out
19:17some models. Well, we know the last
19:20model with all three variables is junk.
19:24The VIFs are sky-high. The R square
19:28predicted is way lower than the R
19:30squared adjusted. So that model is no
19:33good. Now we can rule out also the top
19:36of the two variables. So again there we
19:39have a VIF of
19:4111.59 and the R square predicted falls
19:44off a cliff from
19:4583.47 for the R squ adjusted. That one's
19:49gone. So we can rule out those two right
19:53off the
19:54bat. Now how do we decide? So here is
19:58sort of the golden rule of choosing your
20:00multiple regression model. We want to
20:03look at several factors. We want to look
20:05at the R squar adjusted. We want the
20:09highest one we can get. We want the R
20:11square predicted to be as high as we can
20:14get and to be close to the R squ
20:17adjusted which is already high. We'd
20:20like to see a relatively small standard
20:22error of the regression. So that's the S
20:24column over here on the left. And
20:26finally, all else being equal, we want
20:29the simplest model there is. So if we
20:34look at some other candidates down here,
20:36we can see that for the two variable
20:38model, we have X1, X3, X2, X3. But those
20:44have some serious problems. Remember
20:46that the X3 or the gas price coefficient
20:49was negative and that's because the gas
20:52price
20:53coefficient doesn't contribute to the
20:55dependent variable at all. Plus, we have
20:58some sharp falloff in the R square
21:00predicted. So, we're going to rule those
21:02out. So, we have ruled out all the two
21:04variable options and the three variable
21:08option. So, basically we're at the top.
21:11We can definitely eliminate the bottom
21:14one variable with just x3 in it. We know
21:17x3 is basically a junk variable. So we
21:19can x that out. Now we have to decide
21:24between the top two. That's all we have
21:26left. So is it going to be the top model
21:29with x1 or the next one with just
21:32x2? Well, I think it's pretty obvious
21:35that the top model with just
21:38x1 is the best model.
21:41So a one variable model is the best
21:46model out of all these options. So we
21:50have a very narrow standard error of the
21:52regression at 34 and some change a very
21:55high R squared a very high R square
21:58predicted no multiolinearity problems
22:00because well there's only one variable.
22:02So guess what that is our best
22:08model. So yes, and you are going to kill
22:11me, but Mini Tab and I'm sure other
22:13stats packages can do all of this,
22:15everything we just did by hand looking
22:18at the relationships, it can do it in a
22:20few clicks. And here it is. This is the
22:22output from Mini Tab. It's basically a
22:25best subsets regression which we just
22:27did step by step. So how do we read this
22:29thing? Well, look at the R squared
22:32adjusted. Which are the highest values?
22:35So we have 844. We look we have
22:39855 and we have an 842 in there. So the
22:42855 is the highest. Then we have the 844
22:45at the top. Now look at the R squar
22:48predicted which are the highest values.
22:51So we have
22:5279.1 at the top and then from there they
22:56go down pretty quickly. So nothing
22:59really worth mentioning. The 79.1 is
23:02definitely the highest R square
23:03predicted.
23:05Now examine the difference between the R
23:07squared adjusted and the R square
23:09predicted. A large drop off from the
23:13adjusted to the predicted indicates
23:16overfitting. That indicates there are
23:19too many variables in the model. So we
23:22can see that for the ones there at the
23:24bottom like the three variable model we
23:27go from 84.2 adjusted to
23:3157.5 predicted. That is a sign of
23:35overfitting and it's a bad thing. Now
23:38look at the top. We go from 844 to
23:4179.1. That's very close. And the ones
23:44below it aren't too bad either. But you
23:46can see as we get down with more
23:49variables that the drop off is very
23:51high. So those models are
23:55overfitted. Now look at Maloc.
23:58Look for the one that is low and
24:00approximately equals the number of
24:03predictor variables or independent
24:05variables plus the constant which is
24:08one. There's one constant. So in the
24:10first example we have one independent
24:12variable or one predictor variable plus
24:14the constant. So that's two. So for the
24:17single variable models we're looking for
24:19a malo's number that is two. For two
24:23variables we look for three. That's 2 +
24:261. And then for the three variable, we
24:28look at four. So overall, we're looking
24:30for the lowest one that's closest to its
24:32magic
24:33number. Then using all of the
24:36information we have above, choose the
24:39best model. And based on that info,
24:42which is the best model? The first one.
24:45So the single variable X1 model that has
24:49the very high R squared adjusted the
24:51very high R square predicted that
24:53doesn't fall off the malo CP that is
24:56almost exactly two which is what we want
24:58and a relatively narrow standard error
25:00of the regression. So at 342 it's kind
25:04of in the middle of the pack but it's
25:06fine. So overall that is the best model
25:10and that is the one we would use believe
25:13it or not to make our
25:15predictions. Okay. So that was a tour
25:17day force of how to evaluate multiple
25:21regression models. Yes, that was long. I
25:25admit that. But you'll come out of it
25:28never having to really doubt or question
25:30your knowledge of how multiple
25:32regression works, how the best models
25:34are built, at least in the linear cases.
25:37So when you go to take your test, write
25:39your paper, write a report at work,
25:41you'll be able to create the best
25:43models, make the best predictions on
25:45those models, and substantiate any
25:47findings or suggestions you make,
25:49whether it's in a report or on a paper
25:52or whatever else it might be. So, I know
25:54that was long, so I'll let you go. Thank
25:56you very much for watching. Please
25:57subscribe. If you like the video, give
25:59it a thumbs up. And I look forward to
26:01seeing you again next time.
26:05[Music]

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