analysis of covariance - stern.nyu.edujsimonof/classes/2301/pdf/ancova.pdf · analysis of...

Analysis of covariance

Analysis of variance (ANOVA) models are restrictive in that they allow only categori-

cal predicting variables. Analysis of covariance (ANCOVA) models remove this restriction

by allowing both categorical predictors (often called grouping variables or factors) and

continuous predictors (typically called covariates) in the model. So, for example, in the

mileage of automobiles example, potential predictors of miles per gallon could be size and

year of the auto (grouping variables), and also the weight and engine size of the auto

(covariates).

The standard ANCOVA model incorporates covariates into an ANOVA model in a

straightforward way. If there is one grouping variable, for example, the model is

yij = µ + αi + β1x1ij + · · · + βpxpij + εij ,

where αi is the corrected effect on y given that you are in group i (corrected in the sense

that the covariates x1, . . . ,xp are taken into account). This model is fit using K − 1 effect

codings to represent the grouping variable, along with the p covariates and the constant

term.

From this model, we have several hypotheses of interest:

(1) Are there differences in level between groups (given the covariates)? This tests the

null hypothesis

H0 : α1 = · · · = αK = 0.

The test used for this hypothesis is the partial F–test for the K − 1 effect coding

variables (that is, it is based on the residual sum of squares using all of the variables,

and the residual sum of squares using only the covariates).

(2) Do the covariates have any predictive power for y (given the grouping variable)? This

tests the null hypothesis

H0 : β1 = · · · = βp = 0.

The test used for this hypothesis is the partial F–test for the covariates (that is, it is

based on the residual sum of squares using all of the variables, and the residual sum

of squares using only the effect codings).

c© 2016, Jeffrey S. Simonoff 1

(3) Does the particular variable x` provide any predictive power given the grouping vari-

able and the other covariates? This tests the null hypothesis

H0 : β` = 0.

The test used for this hypothesis is the usual t–test for that covariate.

This model generalizes to more than one grouping variable as well. For two grouping

variables, for example, the model is

yijk = µ + αi + βj + (αβ)ij + γ1x1ijk + · · · + γpxpijk + εijk ,

which allows for two main effects (fit using effect codings for each grouping variable) and

an interaction effect (fit using the pairwise products of the effect codings for the main

effects), as well as the presence of covariates. (I’ve changed the slope coefficients for the

covariates to γ’s so that the earlier (α, β) notation used for two–way ANOVA can be used

here as well.) The usual ANOVA–type hypotheses about the significance of main effects

and the interaction effect are tested using the appropriate partial F–tests.

This regression approach is not always used by statistical packages; some use a cell

means approach instead, which can give different answers (usually only slightly different).

Many ANCOVA routines (including that of Minitab) are quite restrictive, being designed

for use only with balanced designs (and even then only giving approximate F–tests), so

I don’t recommend using them (the Minitab general linear model routine fits the exact

F–tests correctly).

Both models mentioned here are constant shift models, in the sense that the only

differences between the expected value of the target variable between groups is one of shift,

with the slopes of the covariates being the same no matter what group an observations falls

in. Indeed, the ANCOVA model with one grouping variable is identical to the constant shift

model that we’ve used before, except that more than two levels of the grouping variable is

allowed. This leads to a natural question: might the slopes also be different between levels

of the grouping variable? Of course, this is exactly the same as the question of whether

the full model is an improvement over the constant shift model, and it is tested the same


way. Assume for simplicity that there is one covariate x in the data. A generalized model

that allows for different slopes for different groups is

yij = µ + αi + β1ixij + εij ,

where β1i is the slope of x for the ith group. If the interaction of the grouping variable

and the covariate are entered as part of the general linear model, the partial F–test for

this set of variables is a test of the hypothesis

H0 : β11 = · · · = β1K

(this is often called a test of common slope). This is easily generalized to p > 1 covariates

using the appropriate interaction terms. Note that this model can only be fit if you have at

least p + 1 observations within each group (for example, in the simple regression situation

you need at least two observations in each group). This could also be generalized to the

situation with more than one grouping variable, but that is rarely done.

By the way, this is one way of deseasonalizing time series data. Say you have quarterly

data, and you want to regress a target variable on a set of predictors. You might think

that your target might exhibit a time trend, so you include time as a predictor in the

model (detrending). You also think that there might be seasonal effects in the data. You

can include those possible effects (that is, deseasonalize the data) by creating a variable

that defines the four quarters and then include it as a grouping variable in the general

linear model. The partial F–test for that effect is a test of a seasonal effect on level (for

example, the target is higher in the spring given the covariates, lower in the summer, etc.).

Even if this test is not significant, however, you might very well find that a significant

lag–4 autocorrelation (before deseasonalizing) that was indicating a seasonal effect is no

longer significant. For monthly data, you would use a variable defining the 12 months. For

weekly data, you can imagine using a variable with 52 levels, but that would require lots

of data to be reasonable.


Trade breaks on the exchange floor

When a customer calls a stock trading house to place an order to buy or sell stocks

listed on the New York Stock Exchange, the office contacts the trader, who goes to the

specialist booth and says “I want to buy x shares of XYZ at $10”. The trader writes the

order down on a piece of paper (“I bought x shares of XYZ at $10.”), and the person

at the booth also records the trade (“I sold x shares of XYZ at $10.”). This is called

executing the trade. The pieces of paper are later matched up (the matching process). If

the information on the pieces of paper doesn’t match, this is called a trade break. It is

labor intensive to resolve these breaks, as someone has to go back to the people involved

and ask questions, so it is important to the trading house to understand and control trade

breaks. The following data refer to all of the daily trades that occurred from June 1995

through May 1996 at a large New York City investment house (sorry, but I’m not allowed

to say which one). For each day the total number of trades (Trade Total), total number

of trade breaks (Trade Breaks), the percent of the trades the resulted in breaks (Break

Rate), and the day of the week are recorded. Is it possible to build a model that describes

and predicts break rates?

First, here are some descriptive statistics. The break rate is about 7.5%, which is

around the industry average. With an average of almost 3000 trades daily, this translates

into more than 200 trade breaks per day on average.

Descriptive Statistics

Variable N N* Mean Median Tr Mean StDev SE Mean

Total_Br 254 2 219.57 206.50 210.29 92.53 5.81

Trade_To 254 2 2996.6 2952.0 2992.5 695.6 43.6

Break_Ra 254 2 7.545 7.019 7.212 2.815 0.177

Variable Min Max Q1 Q3

Total_Br 79.00 1298.00 180.00 241.00

Trade_To 1057.0 5383.0 2570.2 3437.7

Break_Ra 3.674 28.496 6.141 8.233


A histogram of break rates strongly suggests a long right tail:

3020100

100

90

80

70

60

50

40

30

20

10

0

Break rate

Fre

qu

en

cy

We will therefore work in the log scale, although even for this variable there is a bit

of a long tail:


1.51.41.31.21.11.00.90.80.70.60.5

60

50

40

30

20

10

0

Logged break rate

Fre

qu

en

cy

Side–by–side boxplots give evidence of a day of the week effect on logged break rate,

with the middle days of the week having lower break rate. This suggests the possibility

of some sort of psychological (carelessness) effect related to being close to the weekend

(either looking forward to it or recovering from it!).


FridayThursdayWednesdayTuesdayMonday

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

Day of week

Lo

gg

ed

bre

ak r

ate

There also appears to be a relationship between logged break rate and the total number

of trades, with busier trading days associated with lower break rate. This is puzzling, as

we might expect a higher break rate on busy days. The explanation I was given is that on

days when volume is anticipated to be high, the traders are told to be particularly careful

in recording their trades. Another possibility is that the people who perform the matching

don’t want to work so hard, so on busier days they certify more trades as matched when

they actually weren’t. A third suggestion I’ve gotten is that breaks occur more often in

the morning; if on busy days trading gets heavier in the afternoon (when there is a lower

chance of a break), the percentage of trades broken for the day would go down. A fourth

suggestion is that on high volume days the traders are more likely to leave their slips with

the specialist and let him (or his clerk) write down the details of the trade; since it is all

internal to the specialist booth, there is less likelihood of mistakes. A fifth suggestion is

that on busy days the specialist acts as one side of the trade more often, leading to less

confusion and therefore fewer trade breaks.


1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Total number of trades

Lo

gg

ed

bre

ak r

ate

The regression model that relates the logged break rate to the day of the week and the

total number of trades is an ANCOVA model, with day of the week as a grouping variable

and trade total a covariate. Here is the ANCOVA output:

General Linear Model: Logged break rate versus Trade_Total,

Day of week

Method

Factor coding (-1, 0, +1)

Rows unused 2

Factor Information

Factor Type Levels Values

Day of week Fixed 5 Friday, Monday, Thursday, Tuesday,

Wednesday


Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Trade_Total 1 0.78921 0.789215 70.78 0.000

Day of week 4 0.18560 0.046401 4.16 0.003

Error 248 2.76508 0.011150

Lack-of-Fit 244 2.75395 0.011287 4.06 0.088

Pure Error 4 0.01113 0.002783

Total 253 3.76889

Model Summary

S R-sq R-sq(adj) R-sq(pred)

0.105591 26.63% 25.15% 22.36%

Coefficients

Term Coef SE Coef T-Value P-Value VIF

Constant 1.1004 0.0295 37.33 0.000

Trade_Total -0.000081 0.000010 -8.41 0.000 1.01

Day of week

Friday 0.0079 0.0132 0.60 0.549 1.61

Monday 0.0392 0.0137 2.86 0.005 1.67

Thursday 0.0146 0.0132 1.11 0.268 1.61

Tuesday -0.0350 0.0132 -2.64 0.009 1.62

Regression Equation

Day of week

Monday Logged break rate = 1.1395 -0.000081Trade_Total

Tuesday Logged break rate = 1.0654 -0.000081Trade_Total

Wednesday Logged break rate = 1.0736 -0.000081Trade_Total

Thursday Logged break rate = 1.1150 -0.000081Trade_Total

Friday Logged break rate = 1.1083 -0.000081Trade_Total


Means

Fitted

Term Mean SE Mean

Day of week

Monday 0.8977 0.0155

Tuesday 0.8235 0.0148

Wednesday 0.8318 0.0146

Thursday 0.8731 0.0147

Friday 0.8664 0.0147

Data

Covariate Mean StDev

Trade_Total 2997 696

Both the day of the week and the total number of trades are significant predictors for

logged break rate. The entries under Term show that given the day of the week there is an

inverse relationship between logged break rate and the total number of trades, with 100

additional trades associated with an increase in logged break rate of (100)(−.000081) =

−.0081. That is, 100 more total trades is associated with multiplying the break rate by

10−.0081 = .982, or a reduction of 1.8% (remember that this is a semilog model), given

the day of the week is held fixed. The entries under Means show that given the total

number of trades, break rates are lower in the middle of the week and particularly high

on Mondays (I guess those traders both work hard and play hard!). The difference in

fitted means between Monday and Tuesday, for example, is .8977 − .8235 = .0742, which

means that given that the total number of trades is the same, the expected break rate on

Monday is a multiplicative factor of 10.0742 = 1.186 higher than that on Tuesday (that is,

18.6% higher). We see that about one–quarter of the variability in logged break rates is

accounted for by the model.

The fitted means deserve further comment. The adjustment here refers to the fact

that these are estimated means for the days of the week given the covariate(s) (specifically,

they estimate the expected y if all covariates equal their mean value). These are not the

same as the ordinary means:


Descriptive Statistics

Variable Day of w N Mean Median Tr Mean

Logged b Monday 47 .9079 0.8807 0.8936

Tuesday 51 0.8264 0.8301 0.8228

Wednesda 52 0.8299 0.8203 0.8276

Thursday 52 0.8677 0.8438 0.8632

Friday 52 0.8615 0.8464 0.8506

Note, for example, that the fitted mean for Monday is slightly smaller than the or-

dinary (unadjusted) mean. This is because part of the high break rate on Mondays is

accounted for by the low average number of trades on that day (recall that there is an

inverse relationship between break rate and total number of trades). The overall pattern

is not very different between the adjusted and unadjusted means, however.

Since we are fitting a model that does not include an interaction, we can compare the

fitted means to see which days are significantly different from each other. Here are the

Tukey comparisons:

Comparisons for Logged break rate

Tukey Pairwise Comparisons: Response = Logged break rate,

Term = Day of week

Grouping Information Using the Tukey Method and 95% Confidence

Day of week N Mean Grouping

Monday 47 0.897660 A

Thursday 52 0.873103 A B

Friday 52 0.866389 A B

Wednesday 52 0.831783 B

Tuesday 51 0.823543 B

Means that do not share a letter are significantly different.


Tukey Simultaneous Tests for Differences of Means

Difference of Day of Difference SE of Simultaneous 95%

week Levels of Means Difference CI

Monday - Friday 0.0313 0.0213 (-0.0269, 0.0895)

Thursday - Friday 0.0067 0.0207 (-0.0498, 0.0632)

Tuesday - Friday -0.0428 0.0208 (-0.0997, 0.0140)

Wednesday - Friday -0.0346 0.0207 (-0.0911, 0.0219)

Thursday - Monday -0.0246 0.0213 (-0.0828, 0.0337)

Tuesday - Monday -0.0741 0.0214 (-0.1324, -0.0158)

Wednesday - Monday -0.0659 0.0213 (-0.1240, -0.0077)

Tuesday - Thursday -0.0496 0.0208 (-0.1064, 0.0073)

Wednesday - Thursday -0.0413 0.0207 (-0.0979, 0.0152)

Wednesday - Tuesday 0.0082 0.0208 (-0.0486, 0.0651)

Difference of Day of Adjusted

week Levels T-Value P-Value

Monday - Friday 1.47 0.585

Thursday - Friday 0.32 0.998

Tuesday - Friday -2.06 0.239

Wednesday - Friday -1.67 0.452

Thursday - Monday -1.15 0.779

Tuesday - Monday -3.47 0.005

Wednesday - Monday -3.09 0.017

Tuesday - Thursday -2.38 0.121

Wednesday - Thursday -1.99 0.268

Wednesday - Tuesday 0.40 0.995

Individual confidence level = 99.32%

We see that the important difference is the adjusted rate on Monday being much higher

than that on Tuesday and Wednesday.

Let’s check some assumptions. Here are residual plots and diagnostics (remember, the

model is based on 5 predictors, not 2):


Row Trade_Date Break_Rate SRES1 HI1 COOK1

1 6/1/95 7.5117 -0.15049 0.0199450 0.000077

2 6/2/95 5.5615 -1.30721 0.0197564 0.005740

3 6/5/95 24.8516 4.27566 0.0234331 0.073111

4 6/6/95 8.3578 1.26281 0.0212740 0.005777

5 6/7/95 5.5786 -0.70698 0.0193445 0.001643

6 6/8/95 6.2500 -0.87808 0.0197350 0.002587

7 6/9/95 7.4188 -0.01752 0.0193754 0.000001

8 6/12/95 7.5977 -0.34226 0.0213678 0.000426

9 6/13/95 6.4541 -0.45916 0.0208552 0.000748

10 6/14/95 6.1668 -0.33382 0.0192619 0.000365

11 6/15/95 10.2936 1.09845 0.0203740 0.004182

12 6/16/95 7.0064 -0.33156 0.0196726 0.000368

13 6/19/95 6.4072 -0.94339 0.0212861 0.003226

14 6/20/95 3.6739 -1.72460 0.0280829 0.014323

15 6/21/95 5.6180 -0.62609 0.0195115 0.001300

16 6/22/95 7.0009 0.02173 0.0200130 0.000002

17 6/23/95 6.4358 -0.03521 0.0222988 0.000005

18 6/26/95 7.2753 -0.49159 0.0213116 0.000877

19 6/27/95 6.0967 -0.54122 0.0199018 0.000991

20 6/28/95 8.4848 1.03317 0.0193407 0.003509

21 6/29/95 4.9551 -1.52346 0.0194578 0.007676

22 6/30/95 7.6557 -0.03752 0.0201153 0.000005

23 7/3/95 6.0264 -1.80424 0.0258596 0.014403

24 7/4/95 * * * *

25 7/5/95 10.7401 0.45451 0.0493943 0.001789

26 7/6/95 7.3794 0.08602 0.0193268 0.000024


27 7/7/95 5.5195 -0.41448 0.0266280 0.000783

28 7/10/95 6.6996 0.12529 0.0326899 0.000088

29 7/11/95 7.4722 1.08049 0.0250675 0.005003

30 7/12/95 6.7261 0.33419 0.0209729 0.000399

31 7/13/95 7.1721 0.15689 0.0202585 0.000085

32 7/14/95 7.2886 0.29892 0.0203771 0.000310

33 7/17/95 5.9095 -0.67417 0.0268252 0.002088

34 7/18/95 7.5023 0.71235 0.0204440 0.001765

35 7/19/95 6.0193 -0.09409 0.0213143 0.000032

36 7/20/95 6.4900 0.22193 0.0270591 0.000228

37 7/21/95 8.5550 0.72556 0.0192642 0.001723

38 7/24/95 8.8662 0.50835 0.0215010 0.000946

39 7/25/95 4.5055 -1.04446 0.0247630 0.004617

40 7/26/95 8.7409 0.96215 0.0193848 0.003050

41 7/27/95 5.8233 -0.65490 0.0207067 0.001511

42 7/28/95 22.0989 4.69257 0.0193079 0.072256

43 7/31/95 8.0958 -0.18105 0.0217447 0.000121

44 8/1/95 7.4597 0.50625 0.0196637 0.000857

45 8/2/95 7.4451 0.49641 0.0193565 0.000811

46 8/3/95 6.2999 -0.50681 0.0195314 0.000853

47 8/4/95 7.4570 0.39616 0.0203956 0.000545

48 8/7/95 7.5966 -0.11879 0.0215587 0.000052

49 8/8/95 7.2329 0.11013 0.0201863 0.000042

50 8/9/95 8.7169 0.62115 0.0218620 0.001437

51 8/10/95 8.5432 -0.03684 0.0250434 0.000006

52 8/11/95 8.0487 0.02919 0.0213807 0.000003

53 8/14/95 8.6321 -0.20206 0.0243494 0.000170

54 8/15/95 8.0039 0.03145 0.0264693 0.000004

55 8/16/95 8.0074 0.05020 0.0251470 0.000011

56 8/17/95 12.3293 1.86214 0.0202661 0.011955

57 8/18/95 8.6859 0.44157 0.0204629 0.000679

58 8/21/95 8.5016 -0.36636 0.0257983 0.000592

59 8/22/95 6.5760 -0.32419 0.0204255 0.000365

60 8/23/95 8.1441 0.00695 0.0273799 0.000000

61 8/24/95 8.0936 -0.39647 0.0276871 0.000746

62 8/25/95 8.4267 -0.02419 0.0248673 0.000002

63 8/28/95 8.0421 -0.62905 0.0263003 0.001781

64 8/29/95 7.3583 -0.21122 0.0245366 0.000187

65 8/30/95 5.5086 -0.89676 0.0192611 0.002632

66 8/31/95 6.5125 -0.71184 0.0197598 0.001702

67 9/1/95 8.5026 0.34294 0.0205472 0.000411

68 9/4/95 * * * *

69 9/5/95 7.1502 -0.50961 0.0279074 0.001243

70 9/6/95 7.2825 -0.00752 0.0206224 0.000000


71 9/7/95 13.5938 2.15393 0.0213205 0.016845

72 9/8/95 7.4106 -0.51038 0.0240406 0.001069

73 9/11/95 12.7716 1.14410 0.0292387 0.006571

74 9/12/95 12.2972 1.71666 0.0287036 0.014514

75 9/13/95 8.5791 0.38764 0.0242734 0.000623

76 9/14/95 7.9809 0.22701 0.0193728 0.000170

77 9/15/95 8.2955 0.57064 0.0192375 0.001065

78 9/18/95 8.1206 0.13000 0.0214571 0.000062

79 9/19/95 5.6212 0.18262 0.0311848 0.000179

80 9/20/95 6.7233 0.73810 0.0271811 0.002537

81 9/21/95 6.3502 -0.48152 0.0195007 0.000769

82 9/22/95 8.7002 0.57794 0.0196243 0.001114

83 9/25/95 6.4969 -1.37047 0.0241610 0.007750

84 9/26/95 6.7620 -0.85132 0.0303679 0.003783

85 9/27/95 10.3199 1.45432 0.0205095 0.007381

86 9/28/95 8.0387 0.06790 0.0203987 0.000016

87 9/29/95 8.4517 0.49455 0.0194703 0.000809

88 10/2/95 7.7233 -0.48919 0.0224882 0.000918

89 10/3/95 6.8335 -0.26426 0.0212331 0.000252

90 10/4/95 6.4589 -0.11773 0.0193012 0.000045

91 10/5/95 6.9565 -0.65459 0.0215771 0.001575

92 10/6/95 10.3160 0.75764 0.0259966 0.002553

93 10/9/95 8.9437 0.29957 0.0214676 0.000328

94 10/10/95 9.3417 0.68882 0.0262158 0.002129

95 10/11/95 5.4204 -0.45741 0.0221669 0.000790

96 10/12/95 6.7048 -0.87310 0.0223747 0.002908

97 10/13/95 5.3571 -1.72810 0.0221368 0.011267

98 10/16/95 7.8832 -0.39388 0.0224074 0.000593

99 10/17/95 7.4234 0.10235 0.0210276 0.000037

100 10/18/95 5.1282 -1.64444 0.0226181 0.010430

101 10/19/95 7.4775 0.18104 0.0194417 0.000108

102 10/20/95 5.8842 -0.79786 0.0193195 0.002090

103 10/23/95 10.0042 0.50477 0.0232595 0.001011

104 10/24/95 5.5901 -0.55581 0.0201627 0.001059

105 10/25/95 5.4201 0.30111 0.0396915 0.000625

106 10/26/95 6.4617 -0.20169 0.0209012 0.000145

107 10/27/95 7.0357 0.33553 0.0222988 0.000428

108 10/30/95 9.9240 0.46278 0.0233495 0.000853

109 10/31/95 8.8316 0.26710 0.0302926 0.000371

110 11/1/95 9.8742 0.47450 0.0361008 0.001405

111 11/2/95 15.4913 2.01567 0.0355057 0.024928

112 11/3/95 6.7991 -0.15132 0.0194518 0.000076

113 11/6/95 8.1641 -0.20103 0.0220654 0.000152

114 11/7/95 6.6449 -0.43018 0.0217357 0.000685


115 11/8/95 6.4746 -0.23268 0.0192708 0.000177

116 11/9/95 7.1371 -0.59769 0.0221743 0.001350

117 11/10/95 8.2420 0.20350 0.0206343 0.000145

118 11/13/95 6.2131 -1.34999 0.0221545 0.006882

119 11/14/95 6.4114 -0.80044 0.0247951 0.002715

120 11/15/95 8.4353 0.35584 0.0236488 0.000511

121 11/16/95 5.4218 -0.72494 0.0234843 0.002106

122 11/17/95 6.8269 -0.40459 0.0195174 0.000543

123 11/20/95 8.5468 0.69756 0.0243067 0.002020

124 11/21/95 4.0778 -1.02775 0.0347694 0.006341

125 11/22/95 5.2545 -0.08696 0.0319926 0.000042

126 11/24/95 7.9592 0.28607 0.0193444 0.000269

127 11/27/95 7.0852 -1.93191 0.0466753 0.030456

128 11/28/95 6.9603 0.16512 0.0196093 0.000091

129 11/29/95 6.1395 -0.24113 0.0195790 0.000194

130 11/30/95 6.7538 -0.60398 0.0200197 0.001242

131 12/1/95 7.9251 0.67778 0.0206364 0.001613

132 12/4/95 9.3053 0.78985 0.0218733 0.002325

133 12/5/95 6.4204 -0.22971 0.0196414 0.000176

134 12/6/95 5.1600 -0.65596 0.0222462 0.001632

135 12/7/95 10.1119 1.01601 0.0204426 0.003590

136 12/8/95 9.9586 1.18762 0.0194300 0.004658

137 12/11/95 11.1974 0.82676 0.0251061 0.002934

138 12/12/95 5.9223 -0.75540 0.0203997 0.001981

139 12/13/95 6.2236 -0.47873 0.0194853 0.000759

140 12/14/95 7.3504 -0.57297 0.0235475 0.001320

141 12/15/95 21.3495 3.59935 0.0303446 0.067571

142 12/18/95 10.6968 1.47355 0.0226163 0.008374

143 12/19/95 9.0391 1.66423 0.0220711 0.010418

144 12/20/95 7.1786 0.54289 0.0204190 0.001024

145 12/21/95 8.8826 1.13804 0.0210685 0.004646

146 12/22/95 6.8789 -0.47974 0.0200991 0.000787

147 12/26/95 8.2305 0.25446 0.0245494 0.000272

148 12/27/95 9.8621 0.36100 0.0395668 0.000895

149 12/28/95 11.6165 0.63304 0.0412740 0.002875

150 12/29/95 5.8345 -1.09032 0.0196650 0.003974

151 1/2/96 6.8531 0.01266 0.0196916 0.000001

152 1/3/96 7.0777 0.31401 0.0194405 0.000326

153 1/4/96 10.8424 2.14539 0.0232737 0.018279

154 1/5/96 6.4450 0.15885 0.0252319 0.000109

155 1/8/96 7.7200 -0.04947 0.0215679 0.000009

156 1/9/96 10.5119 0.71959 0.0380747 0.003416

157 1/10/96 10.6657 1.77801 0.0194180 0.010434

158 1/11/96 5.9932 0.38463 0.0405965 0.001043


159 1/12/96 6.9601 0.26580 0.0219852 0.000265

160 1/15/96 5.1998 -0.85079 0.0349039 0.004363

161 1/16/96 6.9606 -0.46828 0.0249666 0.000936

162 1/17/96 7.0323 0.02871 0.0194853 0.000003

163 1/18/96 6.6517 -0.37909 0.0192644 0.000470

164 1/19/96 6.6961 -0.25715 0.0193282 0.000217

165 1/22/96 7.5925 -0.03833 0.0219798 0.000006

166 1/23/96 5.6523 0.24304 0.0321423 0.000327

167 1/24/96 5.1642 -0.69957 0.0216682 0.001807

168 1/25/96 6.3260 -0.78066 0.0195168 0.002022

169 1/26/96 6.4450 -0.58416 0.0193295 0.001121

170 1/29/96 6.7835 -0.87623 0.0215626 0.002820

171 1/30/96 5.6844 -0.24253 0.0223396 0.000224

172 1/31/96 4.7278 -0.79788 0.0258192 0.002812

173 2/1/96 6.8778 0.46784 0.0271397 0.001018

174 2/2/96 7.2654 0.33672 0.0208191 0.000402

175 2/5/96 7.4554 0.12793 0.0242967 0.000068

176 2/6/96 5.1347 -0.64404 0.0226120 0.001599

177 2/7/96 5.5104 -0.23844 0.0244007 0.000237

178 2/8/96 7.4257 0.47159 0.0219395 0.000831

179 2/9/96 6.5992 -0.39878 0.0192309 0.000520

180 2/12/96 5.8960 -0.53340 0.0297784 0.001455

181 2/13/96 6.6022 0.44523 0.0231956 0.000785

182 2/14/96 7.0999 0.43906 0.0199956 0.000656

183 2/15/96 9.3978 1.18074 0.0196481 0.004657

184 2/16/96 6.1224 -0.08522 0.0246955 0.000031

185 2/20/96 6.0702 -0.01575 0.0217995 0.000001

186 2/21/96 4.8283 -0.24549 0.0376897 0.000393

187 2/22/96 5.4516 -0.77447 0.0224433 0.002295

188 2/23/96 4.1906 -1.55897 0.0267064 0.011115

189 2/26/96 8.7662 0.49679 0.0216436 0.000910

190 2/27/96 6.1412 -0.21220 0.0199323 0.000153

191 2/28/96 5.7368 -0.95407 0.0202570 0.003137

192 2/29/96 7.1186 0.66910 0.0283807 0.002179

193 3/1/96 5.9156 -0.61931 0.0200070 0.001305

194 3/4/96 8.6669 0.40531 0.0214720 0.000601

195 3/5/96 8.5890 1.13353 0.0197611 0.004317

196 3/6/96 5.8373 -0.82459 0.0198714 0.002298

197 3/7/96 6.1462 -0.93601 0.0196756 0.002931

198 3/8/96 10.1843 0.71796 0.0257302 0.002269

199 3/11/96 28.4962 6.61720 0.0447372 0.341777

200 3/12/96 6.9865 0.63719 0.0226120 0.001566

201 3/13/96 6.5678 0.09996 0.0198984 0.000034

202 3/14/96 7.7154 -0.37671 0.0236314 0.000572


203 3/15/96 6.3148 -0.56089 0.0192385 0.001029

204 3/18/96 7.3796 -0.64132 0.0222037 0.001557

205 3/19/96 4.6910 -0.71187 0.0279330 0.002427

206 3/20/96 6.6543 0.09794 0.0195997 0.000032

207 3/21/96 6.7989 -0.52222 0.0197029 0.000914

208 3/22/96 5.3612 -1.26312 0.0192308 0.005214

209 3/25/96 7.0688 -0.69018 0.0215072 0.001745

210 3/26/96 5.9172 -0.98018 0.0225274 0.003690

211 3/27/96 5.3770 -0.92691 0.0192384 0.002809

212 3/28/96 5.1291 -1.72086 0.0198557 0.009998

213 3/29/96 5.5121 -1.43737 0.0203752 0.007162

214 4/1/96 6.9712 -0.23125 0.0233557 0.000213

215 4/2/96 7.0710 0.54449 0.0210589 0.001063

216 4/3/96 8.4764 0.76748 0.0196435 0.001967

217 4/4/96 6.8248 -0.57135 0.0200928 0.001116

218 4/8/96 8.6773 -0.27881 0.0257617 0.000343

219 4/9/96 15.3264 1.99476 0.0495235 0.034554

220 4/10/96 9.4247 1.08953 0.0204077 0.004122

221 4/11/96 11.8482 2.77013 0.0278498 0.036639

222 4/12/96 5.4989 -0.26396 0.0303697 0.000364

223 4/15/96 4.8244 -0.37056 0.0654329 0.001602

224 4/16/96 7.3919 0.33320 0.0196788 0.000371

225 4/17/96 7.5973 0.21699 0.0202281 0.000162

226 4/18/96 5.9488 -0.66177 0.0199680 0.001487

227 4/19/96 7.9545 0.39169 0.0192350 0.000501

228 4/22/96 7.5428 -0.52094 0.0220053 0.001018

229 4/23/96 5.2294 -0.30809 0.0269058 0.000437

230 4/24/96 4.6350 -1.23435 0.0207741 0.005387

231 4/25/96 7.8204 0.01142 0.0199794 0.000000

232 4/26/96 8.2421 0.55926 0.0192503 0.001023

233 4/29/96 7.3767 -0.54771 0.0216480 0.001106

234 4/30/96 6.7672 -0.35434 0.0217357 0.000465

235 5/1/96 8.2938 0.79886 0.0192663 0.002089

236 5/2/96 12.6430 1.23278 0.0335079 0.008782

237 5/3/96 6.7494 -0.24352 0.0192885 0.000194

238 5/6/96 7.3204 -0.36315 0.0213146 0.000479

239 5/7/96 6.3079 -0.28776 0.0196236 0.000276

240 5/8/96 6.1533 -0.42551 0.0192479 0.000592

241 5/9/96 4.1923 -1.20468 0.0376266 0.009457

242 5/10/96 6.2698 -0.53657 0.0193144 0.000945

243 5/13/96 6.9206 0.01458 0.0273656 0.000001

244 5/14/96 6.6900 0.12799 0.0198702 0.000055

245 5/15/96 4.7218 -1.25483 0.0200007 0.005356

246 5/16/96 6.2734 -0.35395 0.0206309 0.000440


247 5/17/96 7.3312 0.07593 0.0192537 0.000019

248 5/20/96 6.6265 -1.38201 0.0254509 0.008313

249 5/21/96 4.5416 -1.15233 0.0226419 0.005127

250 5/22/96 6.4129 0.08653 0.0205213 0.000026

251 5/23/96 4.6972 -1.00587 0.0297867 0.005177

252 5/24/96 5.6669 -0.50741 0.0230888 0.001014

253 5/28/96 4.4771 0.19683 0.0680402 0.000471

254 5/29/96 7.0544 -0.09649 0.0202687 0.000032

255 5/30/96 5.8869 -1.32151 0.0212792 0.006328

256 5/31/96 4.2868 -2.46575 0.0202497 0.020944

There are four obvious outliers, each of which has break rate over 20% (they are the

only days with break rates over 20%). What happened on these days? Unfortunately, we

don’t know, but the investment bank should try to track this down. The following output

gives a little clue:

Variable Day of w N Mean Median Tr Mean StDev

SRES1 Monday 47 0.001 -0.231 -0.166 1.380

Tuesday 51 0.001 -0.016 -0.032 0.765

Wednesda 52 0.0002 0.0178 -0.0038 0.7011

Thursday 52 0.001 -0.278 -0.054 0.997

Friday 52 0.000 -0.030 -0.081 1.107

Variable Day of w SE Mean Min Max Q1 Q3

SRES1 Monday 0.201 -1.932 6.617 -0.629 0.405

Tuesday 0.107 -1.725 1.995 -0.510 0.445

Wednesda 0.0972 -1.6444 1.7780 -0.4734 0.4506

Thursday 0.138 -1.721 2.770 -0.660 0.471

Friday 0.153 -2.466 4.693 -0.510 0.395

All of the unusually bad break rate days are Mondays or Fridays, reinforcing that

troublesome weekend effect. We also can note two leverage points (case 127 with a low

number of total trades right after the Thanksgiving weekend and case 253 with a high

number of total trades right after the Memorial Day weekend). The Cook’s distances

are pretty low, so omitting these points probably wouldn’t make much difference, but we

should check; I did so, and the implications of the model did not change.


FridayThursdayWednesdayTuesdayMonday

7

6

5

4

3

2

1

0

-1

-2

-3

C8

SR

ES

1

5500500045004000350030002500200015001000

7

6

5

4

3

2

1

0

-1

-2

-3

Trade_Total

SR

ES

1

The four outliers on Mondays and Fridays are apparent in the side–by–side boxplots.

We can also note that the standard deviation of the residuals is generally higher on Mon-


days and Fridays than in the middle of the week, although the residual plots and a Levene’s

test of heteroscedasticity don’t particularly suggest heteroscedasticity. Note that testing

for nonconstant variance involving the numerical predictor would be based on an ANCOVA

analysis similar to that described in the “CAPM: Do you want fries with that?” handout.

General Linear Model: absres versus Day of week

Method


Rows unused 2

Factor Information



Wednesday



Day of week 4 2.491 0.6227 1.15 0.332

Error 249 134.512 0.5402

Total 253 137.002

Model Summary


0.734989 1.82% 0.24% 0.00%

One other aspect of the data to consider is that it forms a time series. Are there any

autocorrelation effects? There was nothing in the time series plot of the residuals we saw

earlier, but what about tests?


Runs Test

SRES1

K = 0.0005

The observed number of runs = 117

The expected number of runs = 127.0472

116 Observations above K 138 below

The test is significant at 0.2033

Cannot reject at alpha = 0.05

ACF of SRES1

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

+----+----+----+----+----+----+----+----+----+----+

1 0.142 XXXXX

2 0.057 XX

3 0.011 X

4 0.093 XXX

5 0.053 XX

6 0.026 XX

7 -0.014 X

8 0.030 XX

9 -0.067 XXX

The t–test for the lag–one autocorrelation is 2.26, and the runs test is not significant,

so there is not much evidence here of autocorrelation.

It’s possible that heteroscedasticity has had an effect on the analysis, so let’s try a

WLS analysis to be sure. The standard deviations of the residuals separated by day of the

week given above give us weights (one over the variance for each group) that lead to this

output:



Day of week

Method


Weights wt

Rows unused 2

Factor Information



Wednesday



Trade_Total 1 1.27758 1.27758 117.85 0.000

Day of week 4 0.18769 0.04692 4.33 0.002

Error 248 2.68860 0.01084

Lack-of-Fit 244 2.66979 0.01094 2.33 0.213

Pure Error 4 0.01881 0.00470

Total 253 4.15642

Model Summary


0.104121 35.31% 34.01% 31.99%

Coefficients


Constant 1.1295 0.0258 43.72 0.000

Trade_Total -0.000090 0.000008 -10.86 0.000 1.01

Day of week

Friday 0.0085 0.0141 0.60 0.547 2.48

Monday 0.0379 0.0176 2.16 0.032 3.13

Thursday 0.0153 0.0130 1.17 0.242 2.33

Tuesday -0.0353 0.0109 -3.22 0.001 2.12


Regression Equation

Day of week






Means

Fitted

Term Mean SE Mean

Day of week

Monday 0.8964 0.0210

Tuesday 0.8232 0.0112

Wednesday 0.8320 0.0101

Thursday 0.8738 0.0144

Friday 0.8670 0.0160

Data



Both effects are more significant, but nothing very substantive has changed, including

the diagnostics:


Row Trade_Date FITS1 SRES1 HI1

1 6/1/95 0.89434 -0.18093 0.0197882

2 6/2/95 0.88431 -1.21895 0.0195635

3 6/5/95 0.95418 3.10498 0.0221551

4 6/6/95 0.78580 1.73005 0.0218166

5 6/7/95 0.81931 -1.00685 0.0194103

6 6/8/95 0.89008 -0.91652 0.0196243

7 6/9/95 0.87345 -0.02734 0.0193223

8 6/12/95 0.91746 -0.25871 0.0213138

9 6/13/95 0.86160 -0.65685 0.0212614

10 6/14/95 0.82437 -0.47459 0.0192799

11 6/15/95 0.90139 1.08190 0.0201230

12 6/16/95 0.88241 -0.32346 0.0195105

13 6/19/95 0.90488 -0.69095 0.0212805

14 6/20/95 0.73478 -2.16363 0.0308429

15 6/21/95 0.81325 -0.88085 0.0196738

16 6/22/95 0.83988 0.05127 0.0198413

17 6/23/95 0.80634 0.01983 0.0211731

18 6/26/95 0.91384 -0.36572 0.0212909

19 6/27/95 0.84351 -0.74081 0.0199976

20 6/28/95 0.81949 1.51005 0.0194043

21 6/29/95 0.85273 -1.53388 0.0194080

22 6/30/95 0.89109 -0.06230 0.0197907

23 7/3/95 0.97535 -1.37514 0.0231436

24 7/4/95 * * *

25 7/5/95 1.00283 0.39955 0.0668385

26 7/6/95 0.85797 0.09771 0.0193057


27 7/7/95 0.77586 -0.29822 0.0239139

28 7/10/95 0.80159 0.17250 0.0259261

29 7/11/95 0.75288 1.53446 0.0268455

30 7/12/95 0.78837 0.54563 0.0219804

31 7/13/95 0.83581 0.19301 0.0200329

32 7/14/95 0.82778 0.30558 0.0199565

33 7/17/95 0.83379 -0.43836 0.0235370

34 7/18/95 0.79765 0.98379 0.0207163

35 7/19/95 0.78448 -0.06838 0.0225192

36 7/20/95 0.77964 0.31809 0.0253407

37 7/21/95 0.85573 0.67011 0.0192519

38 7/24/95 0.89303 0.38484 0.0213680

39 7/25/95 0.75496 -1.28786 0.0264419

40 7/26/95 0.84228 1.37339 0.0194739

41 7/27/95 0.82948 -0.62595 0.0203827

42 7/28/95 0.85274 4.30706 0.0192796

43 7/31/95 0.92949 -0.14935 0.0214673

44 8/1/95 0.81900 0.68128 0.0196819

45 8/2/95 0.81876 0.73469 0.0194292

46 8/3/95 0.85047 -0.49741 0.0194654

47 8/4/95 0.82751 0.39490 0.0199682

48 8/7/95 0.89123 -0.07462 0.0213915

49 8/8/95 0.85038 0.11331 0.0203747

50 8/9/95 0.88099 0.82296 0.0233838

51 8/10/95 0.94363 -0.11713 0.0237674

52 8/11/95 0.90765 -0.01684 0.0205918

53 8/14/95 0.96313 -0.19017 0.0225284

54 8/15/95 0.90890 -0.07136 0.0287038

55 8/16/95 0.90650 -0.04186 0.0285684

56 8/17/95 0.89976 1.86034 0.0200388

57 8/18/95 0.89643 0.37146 0.0200108

58 8/21/95 0.97489 -0.31961 0.0231186

59 8/22/95 0.85490 -0.46870 0.0206917

60 8/23/95 0.91980 -0.12466 0.0320927

61 8/24/95 0.95928 -0.49911 0.0258308

62 8/25/95 0.93623 -0.09277 0.0227992

63 8/28/95 0.97851 -0.51507 0.0233231

64 8/29/95 0.89633 -0.37597 0.0261417

65 8/30/95 0.83541 -1.30531 0.0192786

66 8/31/95 0.89063 -0.74800 0.0196436

67 9/1/95 0.89761 0.28000 0.0200642

68 9/4/95 * * *

69 9/5/95 0.91714 -0.80103 0.0306102

70 9/6/95 0.86706 -0.06626 0.0214272


71 9/7/95 0.91324 2.14271 0.0208618

72 9/8/95 0.93053 -0.53237 0.0222758

73 9/11/95 0.99678 0.77133 0.0245202

74 9/12/95 0.92139 2.14874 0.0316658

75 9/13/95 0.90062 0.45583 0.0271896

76 9/14/95 0.87959 0.21848 0.0193416

77 9/15/95 0.85889 0.52521 0.0192351

78 9/18/95 0.89457 0.10566 0.0213501

79 9/19/95 0.71932 0.38996 0.0349550

80 9/20/95 0.74116 1.20323 0.0317789

81 9/21/95 0.85137 -0.47264 0.0194415

82 9/22/95 0.88123 0.51079 0.0194799

83 9/25/95 0.96142 -1.04676 0.0224516

84 9/26/95 0.92971 -1.27262 0.0338720

85 9/27/95 0.86553 2.05136 0.0212491

86 9/28/95 0.90175 0.03341 0.0201423

87 9/29/95 0.87689 0.43851 0.0193824

88 10/2/95 0.94260 -0.38562 0.0217702

89 10/3/95 0.86657 -0.40527 0.0217624

90 10/4/95 0.82157 -0.15785 0.0193420

91 10/5/95 0.91595 -0.71621 0.0210621

92 10/6/95 0.94337 0.61578 0.0235141

93 10/9/95 0.92171 0.20971 0.0213544

94 10/10/95 0.90737 0.80317 0.0283678

95 10/11/95 0.77598 -0.58167 0.0238649

96 10/12/95 0.92355 -0.94631 0.0216846

97 10/13/95 0.91515 -1.63292 0.0210705

98 10/16/95 0.94143 -0.31471 0.0217373

99 10/17/95 0.86395 0.08445 0.0214899

100 10/18/95 0.88787 -2.46757 0.0245771

101 10/19/95 0.85327 0.19928 0.0193954

102 10/20/95 0.85211 -0.72207 0.0192869

103 10/23/95 0.95228 0.33712 0.0220844

104 10/24/95 0.80299 -0.70485 0.0203434

105 10/25/95 0.68752 0.65393 0.0515244

106 10/26/95 0.82704 -0.16248 0.0205345

107 10/27/95 0.80634 0.35927 0.0211731

108 10/30/95 0.95328 0.30552 0.0221211

109 10/31/95 0.92935 0.21321 0.0337722

110 11/1/95 0.95923 0.49461 0.0458571

111 11/2/95 0.99474 1.91261 0.0319331

112 11/3/95 0.84668 -0.12465 0.0193707

113 11/6/95 0.93591 -0.16888 0.0215979

114 11/7/95 0.87236 -0.63326 0.0224287


115 11/8/95 0.83622 -0.34595 0.0192939

116 11/9/95 0.92174 -0.66439 0.0215281

117 11/10/95 0.89878 0.15118 0.0201193

118 11/13/95 0.93745 -1.01415 0.0216342

119 11/14/95 0.89814 -1.16025 0.0264843

120 11/15/95 0.89610 0.41649 0.0262039

121 11/16/95 0.80280 -0.66892 0.0225506

122 11/17/95 0.87834 -0.38652 0.0194122

123 11/20/95 0.85315 0.55368 0.0225110

124 11/21/95 0.70385 -1.19684 0.0397069

125 11/22/95 0.71746 0.04298 0.0393731

126 11/24/95 0.87210 0.25205 0.0193027

127 11/27/95 1.06661 -1.52945 0.0316235

128 11/28/95 0.82524 0.22053 0.0196098

129 11/29/95 0.81135 -0.32115 0.0197804

130 11/30/95 0.89569 -0.64359 0.0198465

131 12/1/95 0.82416 0.65598 0.0201206

132 12/4/95 0.88363 0.59876 0.0215197

133 12/5/95 0.83220 -0.31239 0.0196523

134 12/6/95 0.77526 -0.86806 0.0239900

135 12/7/95 0.90239 0.99701 0.0201766

136 12/8/95 0.87554 1.07468 0.0193569

137 12/11/95 0.96956 0.56015 0.0228367

138 12/12/95 0.85445 -1.03980 0.0206576

139 12/13/95 0.84581 -0.71618 0.0196325

140 12/14/95 0.93314 -0.65115 0.0225999

141 12/15/95 0.96644 3.19110 0.0262668

142 12/18/95 0.87151 1.11004 0.0218224

143 12/19/95 0.77703 2.27466 0.0228733

144 12/20/95 0.79561 0.83671 0.0211061

145 12/21/95 0.82505 1.20208 0.0206651

146 12/22/95 0.89082 -0.46710 0.0197805

147 12/26/95 0.89642 0.24174 0.0261587

148 12/27/95 0.97190 0.31041 0.0513275

149 12/28/95 1.01554 0.48611 0.0364352

150 12/29/95 0.88223 -1.01833 0.0195056

151 1/2/96 0.83555 0.00434 0.0197189

152 1/3/96 0.81551 0.47571 0.0195618

153 1/4/96 0.80443 2.24765 0.0223862

154 1/5/96 0.78436 0.21824 0.0230300

155 1/8/96 0.89095 -0.02348 0.0213953

156 1/9/96 0.96173 0.76987 0.0440887

157 1/10/96 0.84355 2.55170 0.0195262

158 1/11/96 0.72221 0.54400 0.0359065


159 1/12/96 0.80923 0.29270 0.0209745

160 1/15/96 0.79173 -0.53435 0.0268281

161 1/16/96 0.89932 -0.72119 0.0267117

162 1/17/96 0.84581 0.01788 0.0196325

163 1/18/96 0.86195 -0.37955 0.0192570

164 1/19/96 0.85166 -0.22630 0.0192925

165 1/22/96 0.88155 -0.00816 0.0215631

166 1/23/96 0.71498 0.47633 0.0362242

167 1/24/96 0.78077 -0.93929 0.0230778

168 1/25/96 0.88457 -0.81168 0.0194540

169 1/26/96 0.87137 -0.54447 0.0192933

170 1/29/96 0.92478 -0.65661 0.0213931

171 1/30/96 0.77440 -0.25049 0.0232292

172 1/31/96 0.74912 -1.03538 0.0296295

173 2/1/96 0.77919 0.56847 0.0254036

174 2/2/96 0.82181 0.34583 0.0202363

175 2/5/96 0.85324 0.13539 0.0225069

176 2/6/96 0.77187 -0.77956 0.0235903

177 2/7/96 0.75834 -0.23838 0.0273906

178 2/8/96 0.81591 0.53387 0.0213449

179 2/9/96 0.86115 -0.36501 0.0192308

180 2/12/96 0.81615 -0.32132 0.0247401

181 2/13/96 0.76680 0.67218 0.0243640

182 2/14/96 0.80239 0.67626 0.0204380

183 2/15/96 0.84739 1.22227 0.0195565

184 2/16/96 0.78789 -0.00845 0.0226904

185 2/20/96 0.77983 0.04283 0.0225132

186 2/21/96 0.69466 -0.15257 0.0483649

187 2/22/96 0.81130 -0.72831 0.0217381

188 2/23/96 0.77541 -1.34477 0.0239635

189 2/26/96 0.88887 0.37948 0.0214261

190 2/27/96 0.80850 -0.25683 0.0200380

191 2/28/96 0.86182 -1.42799 0.0208505

192 2/29/96 0.77250 0.77999 0.0263722

193 3/1/96 0.83375 -0.54110 0.0197222

194 3/4/96 0.89403 0.30838 0.0213562

195 3/5/96 0.81411 1.51953 0.0198111

196 3/6/96 0.85512 -1.23053 0.0202420

197 3/7/96 0.88873 -0.97404 0.0195780

198 3/8/96 0.94175 0.58105 0.0233454

199 3/11/96 0.75546 4.94386 0.0308339

200 3/12/96 0.77187 0.91976 0.0235903

201 3/13/96 0.80420 0.18295 0.0202846

202 3/14/96 0.93377 -0.45228 0.0226654


203 3/15/96 0.85871 -0.51122 0.0192357

204 3/18/96 0.93826 -0.49412 0.0216543

205 3/19/96 0.73560 -0.82029 0.0306442

206 3/20/96 0.81080 0.17016 0.0198130

207 3/21/96 0.88936 -0.55382 0.0195992

208 3/22/96 0.86133 -1.15701 0.0192308

209 3/25/96 0.92306 -0.51861 0.0213705

210 3/26/96 0.88023 -1.37359 0.0234782

211 3/27/96 0.82718 -1.33665 0.0192428

212 3/28/96 0.89262 -1.77636 0.0197185

213 3/29/96 0.89516 -1.34827 0.0199553

214 4/1/96 0.86255 -0.13543 0.0221236

215 4/2/96 0.78851 0.77380 0.0215315

216 4/3/96 0.85015 1.08014 0.0198821

217 4/4/96 0.89696 -0.61173 0.0199035

218 4/8/96 0.97462 -0.25516 0.0231037

219 4/9/96 0.99863 2.41804 0.0592658

220 4/10/96 0.86408 1.52560 0.0210883

221 4/11/96 0.77530 2.91210 0.0259578

222 4/12/96 0.75641 -0.14184 0.0262826

223 4/15/96 0.69874 -0.10865 0.0392649

224 4/16/96 0.83482 0.43027 0.0197019

225 4/17/96 0.86137 0.26709 0.0208049

226 4/18/96 0.84070 -0.64477 0.0198061

227 4/19/96 0.85944 0.36076 0.0192335

228 4/22/96 0.93482 -0.40310 0.0215735

229 4/23/96 0.74139 -0.29223 0.0292824

230 4/24/96 0.79081 -1.72795 0.0216667

231 4/25/96 0.89497 -0.01692 0.0198151

232 4/26/96 0.85708 0.51648 0.0192431

233 4/29/96 0.92713 -0.41699 0.0214279

234 4/30/96 0.87236 -0.53269 0.0224287

235 5/1/96 0.83586 1.14670 0.0192868

236 5/2/96 0.98669 1.12663 0.0303739

237 5/3/96 0.85392 -0.21595 0.0192673

238 5/6/96 0.90181 -0.26222 0.0212921

239 5/7/96 0.83039 -0.38687 0.0196288

240 5/8/96 0.83405 -0.62163 0.0192579

241 5/9/96 0.73270 -1.08035 0.0335885

242 5/10/96 0.85238 -0.48295 0.0192837

243 5/13/96 0.83026 0.06961 0.0237571

244 5/14/96 0.81031 0.19170 0.0199556

245 5/15/96 0.80230 -1.77432 0.0204459

246 5/16/96 0.83048 -0.32094 0.0203235


247 5/17/96 0.85672 0.07405 0.0192453

248 5/20/96 0.97227 -1.06308 0.0229771

249 5/21/96 0.77160 -1.45335 0.0236300

250 5/22/96 0.79416 0.17853 0.0212677

251 5/23/96 0.76544 -0.91435 0.0274695

252 5/24/96 0.79965 -0.40611 0.0216732

253 5/28/96 0.60734 0.57258 0.0838128

254 5/29/96 0.86200 -0.18743 0.0208689

255 5/30/96 0.91279 -1.39113 0.0208295

256 5/31/96 0.89326 -2.28843 0.0198759

Those same four days still show up as outliers, so let’s just see if omitting them changes

things (this is still a weighted analysis):


Day of week

Method


Weights wt

Factor Information



Wednesday



Trade_Total 1 1.29072 1.29072 157.66 0.000

Day of week 4 0.11678 0.02919 3.57 0.008

Error 244 1.99751 0.00819

Lack-of-Fit 240 1.97870 0.00824 1.75 0.316

Pure Error 4 0.01881 0.00470

Total 249 3.40282

Model Summary



0.0904794 41.30% 40.10% 38.27%

Coefficients


Constant 1.1249 0.0227 49.66 0.000

Trade_Total -0.000092 0.000007 -12.56 0.000 1.01

Day of week

Friday 0.0000 0.0125 0.00 0.999 2.53

Monday 0.0209 0.0156 1.34 0.182 3.21

Thursday 0.0239 0.0114 2.10 0.037 2.35

Tuesday -0.02683 0.00956 -2.81 0.005 2.14

Regression Equation

Day of week






Means

Term Fitted Mean SE Mean

Day of week

Monday 0.8709 0.0186

Tuesday 0.82314 0.00970

Wednesday 0.83203 0.00880

Thursday 0.8738 0.0125

Friday 0.8500 0.0142

Data




This weakens the day of week effect slightly, but again not much changes.

Just as was true for ANOVA models, we can obtain a prediction interval for a par-

ticular day of the week and total trades combination from Minitab; for example, for a

Monday with 3100 total trades yields

Prediction for Logged break rate

General Linear Model Information

Terms

Trade_Total Day of week

Variable Setting

Trade_Total 3100

Day of week Monday

Fit SE Fit 95% CI 95% PI

0.861391 0.0187077 (0.824542, 0.898240) (0.612701, 1.11008)

Weight = 0.5251

This PI is (4.099, 12.885) in the original scale, translating into a prediction interval of

roughly (127, 399) total breaks, based on the total trade value of 3100.

We have not yet investigated whether different slopes for total trades for each day of

the week would improve the model. Adding the interaction of total trades and day of the

week doesn’t end up helping:


Day of week

Method


Weights wt


Factor Information



Wednesday



Trade_Total 1 0.64276 0.642761 78.13 0.000

Day of week 4 0.01003 0.002508 0.30 0.875

Trade_Total*Day of week 4 0.02318 0.005796 0.70 0.590

Error 240 1.97433 0.008226

Lack-of-Fit 236 1.95552 0.008286 1.76 0.314

Pure Error 4 0.01881 0.004703

Total 249 3.40282

Model Summary


0.0906993 41.98% 39.80% 36.72%

Coefficients


Constant 1.1006 0.0288 38.21 0.000

Trade_Total -0.000083 0.000009 -8.84 0.000 1.67

Day of week

Friday -0.0117 0.0766 -0.15 0.879 94.80

Monday -0.0587 0.0674 -0.87 0.385 59.48

Thursday 0.0253 0.0525 0.48 0.630 49.89

Tuesday 0.0112 0.0403 0.28 0.781 37.89

Trade_Total*Day of week

Friday 0.000003 0.000025 0.13 0.894 95.13

Monday 0.000028 0.000023 1.23 0.219 64.21

Thursday -0.000001 0.000017 -0.06 0.956 50.01

Tuesday -0.000013 0.000013 -0.99 0.323 38.38


Regression Equation

Day of week






If this were the model of choice, there would be different slope terms for each day of the

week, as can be seen in the output. The t–test for each interaction coefficient refers to

whether the coefficient for that level of the grouping variable is significantly different from

the overall coefficient for all groups (the overall coefficient is just the average of the slopes

for all of the groups, and its estimate is given in the output as −0.000083). The missing

coefficient corresponds to the last group that appears in the data. Since effect codings are

used, the coefficients must sum to zero, so it equals the sum of the other groups’ coefficients

multiplied by −1.

The model that includes the interaction effect between the day of the week and the

total number of trades corresponds to separate regression lines for each day of the week.

Since we only have one numerical variable, we can actually represent that graphically with

separate lines on the same plot:


It is not surprising from these plots that a model with different slopes is not significantly

better than a model with the same slope, although there is a suggestion that Monday might

be different from the other days. Note that an ANOVA interaction plot is not appropriate

in this context of trying to represent a different slopes effect.

So, what have we learned? Most importantly, perhaps, there are real differences in

break rate based on the day of the week. These differences seem important, as they

represent 10–20% differences in break rate between the middle of the week and the ends

of the week. Since trades come in on all days, further investigation of how to improve


performance on Mondays, Thursdays and Fridays seems warranted.

Minitab commands

By default, side–by–side boxplots are given with the boxes ordered either numerically

or alphabetically, as appropriate. This is fine if the categories are defined numerically, but

might not be if they are identified by text. If the default plot doesn’t put the boxes in the

right order, create a variable that defines them numerically, with the numbers assigned

corresponding to the appropriate ordering of the categories (this is done by clicking on

Data → Code → Text to Numeric). Create the side-by-side boxplots in the usual way,

based on the numerical grouping variable just created. Double click on any of the numeric

values that labels for the boxes (below the horizontal axis). Click on the Labels tab and

then the radio button next to Specified, and replace the numeric values in the dialog box

with the text labels given in the correct order.

An analysis of covariance is conducted by clicking on Stat → ANOVA → General Lin-

ear Model → Fit General Linear Model. Enter the target variable under Responses:,

the categorical predictor(s) under Factors:, and the numerical predictor(s) under Co-

variates:. Fitted means, residual plots and storage are obtained as stated earlier in the

ANOVA-related handouts.

Multiple comparisons for categorical predictor(s) in a constant shift ANCOVA model

are obtained in the same ways as are discussed for ANOVA models in the handouts related

to those models.

A Levene’s test when fitting an ANCOVA model can be defined based on both the

categorical and numerical variable(s) using ANCOVA fit to the absolute residuals, although

in that case you probably don’t need to include an interaction effect in the model (note that

this was not done in this handout, where heteroscedasticity was only modeled as a function

of the categorical predictor, day of week). A weighted analysis is obtained by entering a

weight variable under Options. Note that if the observed heteroscedasticity appears to

be related to the numerical variable in the Levene’s test ANCOVA model (as well as the

categorical variable, perhaps), weights should be obtained by saving the residuals from


the original ANCOVA fit, forming the log(residuals2) variable, constructing an ANCOVA

fit with that variable as the target, saving the fitted values from that fit, and setting the

weight variable to 1/exp(FITS). See the “CAPM: Do you want fries with that?” handout

for an example of this.

To fit a model with different slopes for each group, add the interaction effect of the

categorizing variable(s) with the covariate(s) using the same method as was done in two-

way ANOVA. Minitab uses effect codings to fit the models, and as a result estimated slopes

and t-statistics will only be presented for K−1 of the groups, with the coefficient for the last

group (alphabetically) being left out (recall that these coefficients correspond to deviations

from the overall coefficient given earlier in the output). The estimated coefficient for the

omitted group is simply the negative of the sum of the other coefficients (since they must

sum to zero), but if you want to obtain a t-statistic for that slope you must rename the

group so that it is no longer the last one alphabetically.

Remember, an interaction plot is not appropriate for interactions involving covariates;

a scatter plot with different regression lines superimposed is. To construct such a plot,

click on Scatterplots and then With Regression and Groups. Enter the target variable

under Y variables, the predictor under X variables, and the variable that defines the

groups under Categorical variables for grouping. To delete the data points from

the plot (leaving only the regression lines), right click on the plot, then click Select →

Symbols, and press the Delete key.


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