Hi friends welcome to the course of Marketing
Research and Analysis, the second part, so in the last few lectures in fact we have been
discussing about some of the experimental designs and the various techniques in world
in experimental designs such as the ANOVA and ANCOVA, MANOVA. These things are they look very you know very
complex and tough but because thanks to technology and the use of softwares things are become
much easier now a days. And it has been simple and everybody can utilize
them and very nice research papers and publish good research papers with the help of such
techniques, so today we will be; in last the lecture we have discussed about analysis of
covariance where we discussed we said that is a condition where is just like a analysis
of variance but only think is that there is an extra element which is the covariate. So in the ANOVA you had a dependent continuous
dependent variable and a categorical independent variable which is discrete independent variable
but plus in this case there is a covariate which is a extraneous variable and confounding
variable which is a you know measured in the continuous scale. So, this covariate is basically required that
there should be some there should be correlation between the covariate and the dependent variable
but we do not want that there should be any correlation between the independent variable
and the covariate. So when the covariate is present so how when
you control this covariate at different levels of the independent variable what happens to
the dependent variable that is what we measure in case of ANCOVA. ANCOVA is very powerful technique. So it helps you to even measuring interaction
effects with a controlled variable which is the covariate. So today we will start with this examples
so let us go with the problem, A researcher was interested in determining whether a six-week
low-or high-intensity exercise program was good at reducing the blood cholesterol concentration
in middle-aged men, so this is a very general problem, both exercise programs high and low
intensity were designed so number of calories was expended in the low and high intensity
groups. As such, the duration of exercise differed
between groups. The researcher expected that any reduction
in the cholesterol concentration elicited by the interventions would also depend on
the participants initial cholesterol concentration that means what whatever change in the cholesterol
is happening that is not necessarily only due to the exercise program but also due to
the fact with the another factor as to be measured and taken to consideration. That what was is earlier you know cholesterol
level we cannot expect miracles to happen, so if somebody had a very high cholesterol
level before you do not expect that because of just an exercise program drastically reduces
it and very, very high reduction is there, but if that is why the earlier value the pre
exercise value is also important to know ok. As such, the researcher wanted to use pre-intervention
cholesterol concentration as a covariate. So before what was it so that is a covariate
when comparing the post-intervention intervention is the exercise here ok do not confuse with
the words it is very simple intervention cholesterol concentration between the interventions and
a control group. It is a clear case of one way ANCOVA so you
have one covariate one dependent variable which is the blood cholesterol concentration
post exercise and what is the independent variable the exercise the duration of the
exercise intensity of high or low, this is my two groups. So a One way ANCOVA with the dependent variable
which the post which is the post-intervention cholesterol the independent variable is the
group which has three categories so we have three categories are here, so one is no exercise
which is the control group so we are saying let us not give any exercise to somebody intervention
one is the low-intensity exercise and intervention 2 is the high-intensity exercise. So one group; so let us say there are 30 peoples,
so 10 peoples are ask not to do any exercise another 10 people are given a mild exercise
and another 10 people given heavy intensity exercise and pre, which represents the pre-intervention
cholesterol concentrations that means before the exercise of program what was the cholesterol
you know level of all these three groups of people you know these people, so how do conduct
this One way ANCOVA in SPSS. So the simple is you go to analyse and in
analyse there is a general linear model, so these techniques ANOVA sorry Two way, N-way
ANOVA, ANCOVA, one factor, more than one factor MANOVA, MANCOVA they are all coming under
the general linear model techniques, so here, see is there is only the one dependent variable
we say it is a univariate technique. So once you go to the univariate then next
is you have to take here a variables in to the dependent, independent and the covariate
box. So, here we say the group is my fixed factor,
so this is what the treatment we are giving and my pre what was my pre cholesterol level
is my covariate because that is what is effecting my condition my cholesterol condition and
the post exercise is my dependent variable. So after doing my exercise what is the result
of my cholesterol that comes under the dependent variable my group is my this is my fix factor
what group you come in to so no exercise or little exercise or mild exercise and heavy
exercise ok and the pre cholesterol level is taken as my covariate ok. The next is after doing this we what we do
is, we will check for the you know this option where we are trying to display the means,
so here you see if you go back transferred the dependent variable in to the dependent
variable box the independent variable group in to the fix factor and the covariate in
to the covariate box by selecting each variable and clicking the relevant button, so this
is how it goes and then we come to the options button you see this options, this option so
you come to the options button so you get this kind of a screen. So, here we will take check for few things,
so what are few things, the first we will try to see the means for the group, so group
are 3 groups no exercise, light and heavy exercise and we want to see the compare the
main effects so what is the main effects means the effect of exercise you want to see that
bonferroni is taken because the small sample sizes whenever the conditions are there. So you use the bonferroni test, otherwise
you can to check for LSD and other test are there, descriptive, effect of size and effect
size and the homogeneity of variance. These 3 things also required and we are checking
at 5% level of significance you can change that 1% also .001 also it is up to you. Now what is the output so in this case I will
show you first before we go to that. So let me show you let me take this case I
have brought the data set Let us start go to analyze, go to general
linear model, go to univariate, so you can let us do it again so exercise we take it
to the fixed factor, my post cholesterol is my dependent variable my pre cholesterol is
my covariate ok. So now what I am going to do is, I am going
to you see there are two things here you want to check here full factorial model, this is
by default this full factorial model actually is the model which helps you to test whether
the effects of your covariate and the independent variables are they significant or not that
means the different levels what is the change in the dependent variable or is that significant
change or not. So, continue and then we go to options here
we want to see the impact of the exercise that display the means for exercise that means
for exercise for when no exercise is there and there is high level of exercise or the
low level of exercise, now you want to compare the main effects so to compare the main effects
you can take this bonferroni test, now you want to check descriptive statistics we want
to estimates of effect size and homogeneity you when you can check the observed power,
so these four; now let us see what is coming. So when we have done it now we will see there
are three you know groups of exercises so 15, 15 people and the means for the first
exercise group is a 6.11 then 5.57 and 5.41 right now equality of variance levene’s
test is saying that were it is say .301 means null hypothesis which is said that levene’s
test say the homogeneity of variance that means the group should have equal variance
otherwise they should not be compared. So this null hypothesis has been accepted
in this case because it is greater than .05, had it been less than 0.5 you could have rejected
the null hypothesis and said well the variance among the groups is not same and not equal. Then let us come to the test of you know between
subject effects, now you see, we want to see the effect of exercise, now is exercise having
the significant effect on the cholesterol level well the F-ratio says if you look at
the F-ratio what it is 20.813 and it is significant at .001 level and if I see the Partial Eta
Squared, Partial Eta Squared is nothing but my effects size. So, what is the effect size it tells us, it
is like the you know the Eta squared is nothing but similar to the R squared that you talk
about the regression, how much the variance is being explained by the independent variable
so and the observed power of the test is almost 1 so it ranges between 0 and 1 so if it is
1 it is very strong. So, you see in the R squared adjusted R squared
as per is .694 so the Eta squared is something very close. As I said, it explains similar to the R squared
in the regression. So, now we have a test where we can say easily
well the exercise, level of exercise does have an effect on the cholesterol level of
the patients or the people. Now but can we say which cholesterol which
group which exercise is having the highest effect that means no exercise moderate or
high exercise, to do this we have a post hoc test which is the pair comparison. Now let us look at the first 1 and 2, so 1
minus 2 is 1.095 the difference that means it is positive, that means 1 is giving larger
effect than the cholesterol level is more in case of 1 than 2, and is the significant
difference? Yes it is a significant difference. 1 and 3 the differences is 1.624, so again
you see when somebody has done no exercise, the cholesterol level is still higher because
it is the cholesterol level. Higher the cholesterol level it is bad for
the person. So we want actually to be lowest, so in this
case which is lower 1 2, 1 3 so if you see it seems that 3 is the lowest then followed
by 2 and then followed by 1. Let us compare between it is significant,
yes both were significant. Now let us compare between 2 and 3 others
have been compared. So 2 and 3 if you see 2 and 3 it is not coming
significant because the value is .098 that means much higher than .05 which we usually
take for significant testing. By this we can say, well the difference in
the exercises, between that means no exercise and moderate exercise is significant on the
cholesterol level 1 and 3 that is no exercise and heavy exercise is significant, But moderate
exercise which was like you know people walking and hard exercise which is with machines you
know that actually made no significant differences. So this helps us to test the effect of the
treatment on the dependent variable, so this is what we have been finding. But the question is, is it enough? How do we write it now, now see how do we
write? When you write in paper, this is how you show,
so the descriptive statics this is what it shows, it shows that the mean standard deviation
and the number of participants, we saw that these values do not include any adjustments. These are not showing any adjustments, so
what was the purpose of ANCOVA that it helps you to adjust for the change in the dependent
variable, made by the use of a covariate which is helping into that. Now the results if you look at the results. Now look at this, this is the pre, this is
the group so group is the exercise level, low exercise, high exercise and all that. So this table informs whether the different
interventions were statistically significantly different having adjusted for your covariate. The partial Eta Squared this value indicates
the effect size and should be compared with Cohen’s guidelines this is something which
says it is between 0.2 under 0.2 this value, so these two values for example, if it is
a small effect, so the effect is very little. If it is 0.5 then the effect is moderate effect
if it is 0.8 and larger which we is coming to true in both our cases that means the pre
cholesterol levels which is covariate also has the very high effect on my post cholesterol
level and which exercise category I am coming into that also has very high level on the
post cholesterol, But if you have seen comparison that means by previous condition is having
a larger effect even than the exercise that I am doing may be because the duration of
exercise is less. I could extended for let us say not 6 weeks
but 6 months then it could change, and it shows the large effect true similar to the
R squared regression. In order to interpret the results read along
the group, until you reach the significance level, so this, the point we are taking about
taking the point. This provides the statistical significance
of whether there are statistically significant differences in the post intervention blood
pressure or cholesterol whatever you know. I am not a medical person so you can understand
between the groups well adjusted for the blood pressure that is the pre blood pressure that
is the covariate; in this example we see there is the statistical difference, significant
difference between the adjusted means. So it is saying, from here we can say that
the effect is significant ok. How do you report the result it is very very
important because many of times I have seen people do a test but they are not able to
write it. So how do you write, A one-way ANCOVA was
conducted to determine a statistically significant difference between the levels of the independent
variable that is in this case whatever our level of exercise on the dependent variable
which is in this case a post cholesterol. For the pre cholesterol level, so they are
controlling the pre cholesterol level. A one way ANCOVA was conducted to determine
a statistically significant differences between no exercise, low-intensity and high-intensity
exercise on the post intervention cholesterol concentration controlling for pre intervention
cholesterol so the same thing I have written here. Exercise intensity, no exercise, high-intensity,
low-exercise. Post intervention is my dependent variable,
pre intervention is my covariate. A one way ANOVA, this is how you write, was
conducted to check the significant effect of exercise intensity on post-intervention
cholesterol concentration after controlling for the pre intervention cholesterol this
means you are clear, that means what was the previous cholesterol level, if you do not
control it then you just cannot compare because somebody had a high cholesterol level and
somebody had a low cholesterol level. Now you do not expect a similar kind of change
with a same by giving the same exercise to both these people, so that is why you have
to control the pre cholesterol level. Levene’s test and normality assumptions
were carried out and the assumptions met. So you have to check this Levene’s test
homogeneity of variance test all these. There was a significant effect on cholesterol
concentration. So, this is how you write so F at 2 and 41
this is my degrees of freedom the numerator and the denominator is equal to this much
and my p is this much my significant level due to the exercise intensity. Post hoc test which was the paired comparisons
showed there was a significant difference between low-exercise and high, this is 1 and
3 you can say and no exercise and low, which is 1 and 2. Comparing the estimated marginal means showed
that the most cholesterol weight was lost and high exercise with the mean of this much
compared to no and low exercise. So, this is how you report. So, this is estimation of marginal means and
the effect of the finally the F- ratio, we calculate and we report it in the research
paper. So, this is what we talk about in the ANCOVA. Now I will move into another technique which
is the Multivariate Analysis of Covariance, now what is the difference it is just an extension
of ANCOVA, an extension from the ANCOVA. The multivariate analysis of Covariances is
an extension of the MANOVA to incorporate a covariate, so from ANOVA to MANOVA similarly
from ANCOVA to also MANCOVA or an extension of the ANCOVA as it says, to incorporate multiple
dependent variables. In ANCOVA what was happening, you are having
one dependent variable which was continuous then you had a covariate and you had a treatment
level, a treatment factor which had 1, 2 or 3 or 2 or more than 2 levels ok. Now you are saying well it is like a MANOVA
thing, in the MANOVA you have more than one dependent variable so and the presences of
a covariate, so this study now here this is the clear case of MANCOVA. The covariate is linearly related to the dependent
variables the covariate is related to the; we said earlier also the covariate should
be having a some correlation between the, there should be some correlation between the
dependent variable and the covariate. And it is inclusion into the analysis increase
the ability to detect the differences between groups of an independent variable. I am sure why you have understood by now in
the last example also when I explain how the treatment levels that means the exercises
were, effecting the post cholesterol level. But then just saying that and not including
the, what was his pre-cholesterol level will be wrong. So if you take the pre-cholesterol level,
what was is suppose for example somebody is weight how will it change with exercise. But suppose you do not consider the earlier
weight and you think everybody should be coming down to let us say less as 60 kilo but actually
somebody was 90 earlier and somebody was 70 and yourself both of them 60 is wrong. So that is why the covariate is very important. MANCOVA is used to determine whether there
are any statistically significant differences between the adjusted means of 3 or more independent
groups. That means in one factor there are 3 groups
more than that, having controlled for a continuous covariate. You could use MANCOVA to determine whether
a number of different exam performances difference differed based on test anxiety levels amongst
students, while controlling for revision time. So let us see this example that is your dependent
variables would be what is humanities exam performance, science exam performance and
maths exam performance, 3 exams are there, so 3 dependent variables are there, all measured
from 0-100. How much score a student has scored, independent
variable would be the test anxiety level, which has several levels. What are they? 3 groups, low-stressed students, moderately-stressed
students and highly-stressed students, so we are trying to see whether the stress level
of student does it have effect on the exam performance for humanities, science and mathematics
equally or is it differently. So let us see that, but here if you do that
then supposed some student has already done the lot of revision work and somebody has
not done any revision, will their performance are in spite of being you know difference
level of anxieties will the performance be same, no. Because if somebody is well prepared his chances
or her chances of do perform better in the exam at whatever level of stress maybe is
always better. So and your covariate would be the revision
time which is measured in hours. You want to control for revision time because
you believe that the effect of test anxiety levels on overall performance will depend
to some degree on the amount of time spend on revision ok. So assumptions are more or less the same like
ANCOVA. You have 2 or more dependent variable should
be measured at as continuous variables. One independent variable if it is a one way
MANCOVA then it is only one independent variable, Consist of two are more categorical independent
groups. Your one or more covariates all are continuous
ok. So what is a covariate? The covariate is simply a continuous independent
variable that is added to the MANOVA model. MANOVA in the MANOVA there was no covariate
if you add a covariate it become a MANCOVA. There should be a linear relationship between
each pair of dependent variables, so if there are more than one dependent variable there
should be a strong correlation between them. Within each group of the there should be linear
relationship between each pair of dependent variables within each group of the that means
independent variables we are talking about is the factor, levels of the factors ok, levels,
factor levels. So that should be a relationship between the
dependent variables. There should be a linear relationship between
the covariate and each dependent variable there should be within each group of the;
but there should not be any within the covariate and the independent variable. It should be a different level of independent
variable there has to be a correlation between the covariate and the dependent variable,
Simple up to this much. Homogeneity of regression again like ANCOVA
has to be checked ok. Homogeneity of variance and covariance has
to be checked. There should not be significant outliers ok
that was also the same. There should be no significant multivariate
outliers, so well this is both for outliers, so will keep it normally and normality should
be present. And one more thing we missing data should
not be there if it is there you kindly correct. So if it is not normal if it is skewed kindly
transform the data. If there are outliers you may remove them
or you may correct them, if this is not homogeneity of variance is not showing kindly think of
transforming if there is no homogeneity of regression or variance it was the group then
you can think of transforming the data and trained to see if whether it is coming or
not. If still it is not coming then maybe statistical
test, parametric test are not applicable. Now take this example. A Researcher wanted to determine whether cardiovascular
health was better for normal weight individuals with higher levels of physical activity that
is as opposed to more overweight individuals with lower physical activity levels ok. As such the researcher recruited 120 participants
who were subsequently divided into three groups depending on the amount of physical activity. So, each group let us say was the balance
group 40, 40, 40. A group who were classified as engaging in
a low amount of physical activity 40 people, a group who classified as engaging in a moderate
40 people, and high amount of physical activity another 40 people. There 40, 40 on in each group. In order to measure cardiovascular health
the researcher took three measurements from the participants. It was expected that increased levels of physical
activity would have an overall beneficial effect on cardiovascular health, as measured
by cholesterol concentration C-Reactive protein and systolic blood pressure. However the researcher knows that body weight
also effects, the cardiovascular health. So if your body weight is high your chances
are having a cardiovascular problem maybe high. As such the researcher wanted to control for
differences in the body weight of participants. Just understand just imagine had I not taken
the body weight then it would not be justices, would not be the right way of research. You cannot say that any body weight the effect
of exercise of on the you know heart of people would be same for all types of body weights,
no it cannot be true. So what is next, how to conduct the MANOVA,
MANCOVA in SPSS? So here go to general linear model earlier
when we are doing ANCOVA we are going for the univariate. But this time sees we have more than one dependent
variable we go for the multivariate, Then you take all these variables as earlier
we have done. So we are take here let us say cholesterol,
CRP protein and the blood pressure as a dependent variable. The group, which group of activity he was
coming that and weight is my covariate. Now you can similarly I am just showing you
then I will show you is a real case so the same case on the SPSS. So group we will take the group the intensity
of the exercise into the, this side to check the display that means. It is within the multivariate options dialogue
box that you can instruct ok this is. Now take this and finally compare the main
effects and be check for these three things effect size, description statistics, and the
homogeneity. So we want the mean median, standard division
and all those things mean standard division the variance equality and the Eta squared
of effect size, you can also take the observed power to see how stronger test is, how powerful
test is. So as I said if you remembered during ANOCA
if you have a good covariate it reduces the mean sum of square within the test. So that is why if you have a good covariate
it reduces the MS within and that increase the F-ratio so thereby what happens is your
power effect test the beta. The beta reduces the beta is the type to reduces
and your power effect test statistically significantly improve ok. Now click on the continue button and get the
results, so how to interpret the result in the let us try to do it on the SPSS. So I have a MANCOVA. So we go to the general linear model univariate,
so I have done it. We can reset so I will take all the, three
group as my fixed factor, cholesterol, CRP, SBP blood pressure and this has my covariate. Now I will go to straightway I will go to
the full factorial model it is by default; now let us go to or sorry not save I do not
want to save anything in fact so I want to see the group means right group means and
compare the main effects. So you can Bonferroni you can if you keep
LSD does not make much of difference because sample sizes are not large that is why we
are bonferroni actually takes care of that small sample size it adjust for the differences
ok, so but if sample size is good enough then you can take go for even Cedac or LSD and
we want do this right even observed power is want if you can take so this is all; now
run, let us run. So if you go to output now so let us look
at it. so what is the my dependent variable so dependent
variable cholesterol, CRP and SBP and three groups whether High, Low moderate exercise
for all the same and look at the mean and standard deviation. So if you see the cholesterol for the overall
cholesterol was 7.5, CRP 14.34, SBP 128. The highest cholesterol was for the people
who were doing you know high exercise. But here remember we do not know particularly
well because the covariate also is 1 that has to be taken. People with the low-exercise is the mean cholesterol
level is 5.87 and 7.15 this is CRP similarly 36, 2.5 higher the protein risk of you know
heart disease may be high I do not know high or low may be I am not a doctor anyway we
will only go through numbers ok, now coming down so in fact this box’s M tests, we test
the null hypothesis that the observed covariance matrices of the dependent variables are equal
across groups But in this case our data is not showing a
good sign it should not come significant actually please note it down when you write because
of you write in the research paper, we should actually be more than because 5% level significant,
we should have come at least above .05 but our cases coming .000 which show that it is
significant that is actually should not come because I am not treated my data so I have
brought as it is. So, whatever is coming it is shown here. So, you should ensure that it is the above
.05 there is not then we have to may be go to the data set again and check and may be
collect more data my sample size also less here. So, because if you see my sample size how
much it is? It is much less because we expected 120 in
the group but we have only 45 I believe. So you do not expect the same result right
anyway just understand it, now will go to the Multivariate test. Now if you want to see the effect of the weight
so first since it is remember, when it is non-significant you should there are two options,
here basically we go for if it is significant then we should go for the Pillai’s, if it
is significant we go for the Wilks’Lambda, if it is not significant then will go for
the Pillai’s Trace. So here will go for the Wilks’Lambda and
here we say, we see it is significant, so weight has the significant effect right and
the effect size is .387 which is as I said it is close to the similar to the regression
in the R squared in the regression right and observe power is .986, this is the effect
of weight the covariate, now the effect of the group that means the exercise. How much is it? Is it significant? Yes, it is significant and but if you see
the earlier weight of the person has got a larger effect on the final dependent variable
then the type of exercise and observed power is 952 here. So this is Levene’s test should have come
actually this none of them are coming significant but it should have actually for a right MANCOVA
test or ANCOVA test, the Levene’s test should be above .05 or above a significant level,
which is not coming here. Now coming to the test of between subjects
effects now let us look at it we have again see weight we have seen, it is significant? Yes, cholesterol at a particular by controlling
the covariate weight ok. Is cholesterol significant? Yes, is CRP significant? No, CRP is not coming significant. Is the blood pressure coming significant? Yes so when I am controlling for the weight
so my cholesterol is coming significant my blood pressure is coming significant but my
CRP is not necessarily coming significant. When you look at the group so when I look
at the group I will see so group is it significant cholesterol? No, is my CRP significant? No. Is my blood pressure is Significant? Yes that means the type of exercise is having
a effect on these 3. So, obviously my Eta squared if you see my
power effect test will also fall down drastically but how ever would have been higher would
have been more happier, now you can see the pair wise comparisons this is the Post hoc
test, when I am looking at cholesterol. I am comparing cholesterol high and the group
and what I am comparing it against the high cholesterol and I am compare the low and moderate
exercise, so cholesterol low high moderate, cholesterol moderate high and low. So, when I am taking this dependent variables
and I am comparing so I can find out the mean difference from here and I can run the; check
the significance so where then I can say which is having a significant effect which is not
having a significant effect, so but ultimately this is what is the final. So let us go back to the interpreting the
results, now the result values might not be the same here this is only for writing. So the main purpose of running a one-way MANCOVA
is to establish whether the groups of the independent variable are statistically different
on the dependent variables, after controlling for a covariate, if the one-way MANCOVA is
statistically significant, this suggests that there is a statistical significant adjusted
mean difference between the groups of the independent variable in terms of the combined
dependent variable so this is what we saw the last slide. So this table I am taking about ok and look
at this so this is the weight and the group now we will see this group effect for example
we have only interested in the effect of the group so if you look at it is it significant
look at it yes it is significant, so that is what you report and this is my Eta the
effect size. Note: we have highlighted the rows within
the group heading as you saw. The row heading will have the same name as
your independent variable. In our example it is labelled group because
this is the name of our independent variable. So exercise are whatever you want to here,
when you analyze your own data, look for the row heading in the multivariate tests that
matches the name of independent variable. The different names given to each row Pillai’s
Trace, Wilks’Lambda, Hotelling’s and Roy’s are then names of the multivariate statistics
that can be used to test the statistical significance of the differences between groups. But generally we use the Wilks’Lambda, Roy’s
Largest Root is also used but it is a highly you know sensitive to the normality assumption. Each different calculation will provide with
the probability of getting F statistics of greater or equal to one the calculated then
we want to compare with the calculated and the table value, it also provides together
the effect size. The Eta squared so which is analysed, however
you added the complication of having to decide which multivariate F- statistics to use. The most commonly recommended is the Wilks’Lambda
so this is I would have set it I kept it in the power point show that you can later on
read it and understand it. Therefore, the results for the one-way MANOVA
is found along the Wilks’Lambda, so go across this line so what is it saying, is it significant? Yes the Eta squared is this much ok. So, when you look at the Wilks’Lambda that
you ignore all the other things. So when you analyze your data, you will have
an intercept row heading, so these things but since weight reflects the name of our
continuous covariate, weight, this heading name will be different it will have the same
as the name of your continuous covariate. So forget this part this is what we require. If the one way MANCOVA is statistically significant
so what you found, you will have a p-value less than .05. However if p is greater than .05 the one-way
MANCOVA is not statically significant. In our example there is a statistically significant
difference between physical activity groups in terms of the combined health variables,
after controlling for weight alternatively, if p is greater than 0.05 this is not a statistically
significant difference, so between the physical activity groups in terms of the combined health
variables, after controlling for weight .As such, you need to consult the significance
column and along the Wilks’Lambda row. So this is what we have shown earlier also,
so this is written so that is there are statistically significant differences in the combined health
variables between physical activity levels, after controlling for weight. SPSS statistics will also report an effect
size called the Partial Eta Squared, ok. So this is also I have shown, now how do you
finally write so you write like this A One-way MANCOVA was conducted to determine a statistically
significant difference between you have to write the original the name the between here
for the example what is the our independent variable the type of exercise. On the cholesterol, CRP and SBP, so these
are my dependent variables. So you write the levels of the dependent variable
cholesterol, CRP and whatever third one BP blood pressure controlling for what? the weight write so this is have you write. A One-way MANCOVA was conducted to determine
a statistically significant difference between low, medium, and high exercise on the cholesterol
concentration, C-reactive protein, and systolic blood pressure controlling for body weight. There was a statistically significant difference
between the physical activity groups on the combined dependent variables after controlling
for weight. Now this is how you report F(6228) or my numerator,
denominator the degrees of freedom And my F value is this much, p is this much, Wilks’Lambda
is this much, eta squared is this much, so physical activity groups: So, this is the already explain so this is
how finally you see this table. If alternatively the one-way MANCOVA was not
significant you could report the result as follows, suppose sometimes is not significant
so what is your mistake so it is written there was no statistically significant difference
between the physical activity groups on the combined dependent variables after controlling
for weight, this is same write is equal to this much, p is this much, Wilks’Lambda
is this much and eta squared. So you have got that you remember so this
is my final if you so just one you can go back of also and check the table value, so
individually for the groups we had seen we should go to the output table, now if you
go to the output little up also you can it here, so this does this group was the group
effect significant let us say Wilks’Lambda will see is it significant? Yes, so we can see here and then from here
from this you can compare the test of between subject effects, so now will stop here I am
sure you must have got the some insight at least into the details of ANCOVA and MANCOVA. Which is nothing but similar to the extension
of ANOVA and MANOVA, MANOVA is sometimes called as the mirror image of the discriminated analysis
that is the separate story , I will explain sometime, but I hope it is clear to you if
it is there is some doubt in your mind you can question me and I will try to help you
out but remember it is the very very important technique, very important techniques and the
very useful and you can really do wonderful research work if you can understand this statistical
tools, I hope this is all for the day, thank You very much.
