1.A student received an "A" on the first test of the semester. The student wants to calculate the probability of
scoring an "A" on the second test. Historically, the instructor knows that the joint probability of scoring "A"'s on the
first two tests is 0.
M+F
c)
F
M
y
x
b)
F
M
Figure 19.1 Regression models with sex as a categorical independent variable: a) no
difference between males (M) and females (F); b) a difference exists but the slopes are
equal; c) a difference exists and slopes are different
There
of variation are Model, residual (Error) and
Corrected Total. In the table are listed
degrees
of freedom (DF), Sum of Squares, Mean
Square, calculated F (F value) and P value
(Pr>F). In the next table F tests for initial,
treatment and initial * treatment
Cov Parm Subject Estimate
CS kid(treatment) 0.02085
Residual 0.01106
Fit Statistics
-2 Res Log Likelihood -59.9
AIC (smaller is better) -55.9
AICC (smaller is better) -55.7
BIC (smaller is better) -54.3
Null Model Likelihood Ratio Test
DF Chi-Square Pr >
indicating interaction between x1i and x2i. The expectation of the
dependent variable is:
E(yi) = 0 + 1x1i + 2x2i+ 3x1ix2i
Chapter 19 Analysis of Covariance 361
For males (M) the model is:
E(yi) = (0 + 2) + (1 + 3)x1i
For females (F) the model is:
E(yi) =
A 400 1000 B 340 950 C 320 940
A 360 980 B 410 980 C 330 930
A 350 980 B 430 990 C 390 1000
A 340 970 B 390 980 C 420 1000
;
PROC GLM;
CLASS treatment;
MODEL gain = initial treatment / SOLUTION SS1;
LSMEANS treatment / STDERR PDIFF TDIFF ADJUST=TUKEY;
RUN
(*x)ij = interaction of group x covariate
ij = random error
The overall mean is: = 0 + 1x
The mean of group i is: i = 0 + i + 1x + 2ix
The intercept for group i is: 0 + i
The regression coefficient for group i is: 1 + 2i
The hypotheses are the following:
+ ij + tk +(*t)ik + ijk i = 1,.,a; j = 1,.,b; k = 1,.,n
where:
yijk = observation ijk
i
366 Biostatistics for Animal Science
= the overall mean
i
= the effect of treatment i
tk = the effect of period k
(*t)ik = the effect of interaction between treatment
0.00015 0.01127 0.02062 0.02849
0.01127 0.02062 0.02849 0.02062
0.02062 0.02849 0.02062 0.01127
0.02849 0.02062 0.01127 0.00015
SAS gives several criteria for evaluating model fit including Akaike
information criteria
(AIC) and Swarz Bayesian information
subjects is equal to zero.
More complex models can include different variances and covariances
for each
treatment group for both the between and within subjects. These will be
shown using SAS
examples.
20.3.1 SAS Examples for Random Coefficient Regression
that can be tested against zero, that is, if the slope for that group is
different than the average
slope of all groups. This multiple regression model is equivalent to the
model with the
group effect as a categorical variable, a covariate and their inter
340 970 390 980 420 1000
To show the efficiency of including the effect of initial weight in the
model, the model for
the completely randomized design without a covariate is first fitted. The
ANOVA table is:
Source SS df MS F
Treatment 173.333 2 86.667 0.
1.Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided
by the total number of possible outcomes. (Points:1)
True
Question2.2.If only one of several events can occur at a time, we refer to these
1.What is the median of 26, 30, 24, 32, 32, 31, 27 and 29? (Points:1)
29.5
Question2.2.A population consists of all the weights of all defensive tackles on Sociable University's football
team. They are: Johnson, 204 pounds; Patrick, 215 pounds; Junior, 20
Vinod Raman
EMBA CMR 2016B
Economics of IT Quiz 1
1)
a) Owing to health concerns, people start consuming lesser tea than before. For every price
point, the quantity demand falls. This causes the Demand curve to shift left from D1 to D2.
S
P
D1
D2
Q
Ceteri
Vinod Raman, EMBA2016B
Economics of IT
Assignment 2
SINGLE-VERSION
Computations are shown in the sheet titled Single Version.
For each version of Modeler, we consider the Sale Price for each segment and calculate the
profit reaped in each segment. Heres t
yij = observation j in group i (treatment i)
0 = the intercept
1 = the regression coefficient
xij = a continuous independent variable with mean x (covariate)
i = the fixed effect of group or treatment i
ij = random error
The overall mean is: = 0 + 1x
The
It is assumed that covariances between measures on different subjects
are zero.
An equivalent model with a variance-covariance structure between
subjects included the
error term (ijk) can be expressed as:
yijk = + i + tk + (*t)ik + ijk i = 1,.,a; j = 1,.,
The critical value for the model is F0.05,3,10 = 3.71. The null hypotheses if
particular
parameters are equal to zero can be tested using t tests. The parameter
estimates with their
corresponding standard errors and t tests are shown in the following
tabl
UN(1,1) = 0.01673 denotes the variance of measurements taken in
period 1, and
UN(3,1) = 0.01226 denotes the covariance between measures within
animals taken in
periods 1 and 3. The variance-covariance estimates in matrix form are:
0.00792 0.02325 0.03167
B behind the estimates denotes that the corresponding solution is not
unique. Only the
slope (initial) has a unique solution (0.6043956). Under the title Least
Squares Means the
means adjusted for differences in initial weight (LSMEAN) with their
Standard
2
31
23
2
21
12
2
1
32 3
24
2
13 14
where:
i
= variance of measures in period i
ij = covariance within subjects between measures in periods i and j
Another model is called an autoregressive model. It assumes that with
greater distance
between periods, cor
independent. It may be necessary to define an appropriate covariance
structure for such
measurements. Since the experimental unit is an animal and not a single
measurement on
the animal, it is consequently necessary to define the appropriate
experimental
terms are Den DF, and P values are (Pr > F). The P values for fixed
effects in the model
are all smaller than 0.05 indicating that all effects are significant. Note
the different
denominator degrees of freedom (Den DF) indicate that appropriate
errors wer
where:
2 = variance within subjects
2
= covariance between measurements within subjects = variance
between
subjects
This variance covariance structure is called compound symmetry,
because it is diagonally
symmetric and it is a compound of two variances.
i'j
y.y .( )
.
2
ij
i' j
22
'
= + n
s yij yi j
18.2.1 SAS Example: Main Plots in a Completely Randomized
Design
The SAS program for the example of the effect of four pasture
treatments and two mineral
supplements on milk production of cows when pasture t
the experiment. Also, the slopes appear to be different which indicates a
possible interaction
between treatments and initial weight.
800
850
900
950
1000
1050
1100
300 350 400 450
Initial weight (kg)
Daily gain (g/day)
Group A
Group B
Figure 19.2 Daily g
designs the MIXED procedure must be used.
354 Biostatistics for Animal Science
Exercise
18.1. The objective of the study was to test effects of grass species and
stocking density on
the daily gain of Suffolk lambs kept on a pasture. The experiment was
set
week is significant. Also, there is week*treatment interaction, indicating
the effect of each
treatment over time is different. The table Type 1 Test of Fixed Effects
shows that all the
effects in the model are significant.
20.2 Heterogeneous Variances an