The session started with a new statistics application named as PERMAP. Lets have
An Introduction to PERMAP
PERMAP is
a program that uses multidimensional scaling (MDS) to reduce multiple pair wise
relationships to 2-D pictures, commonly called perceptual maps. The Churchill
data are in the form of correlation coefficients that show the relationships
between 10 factors that influence the image of a department store. These
correlation coefficients were calculated from responses to semantic
differential scale questions given to a random selection of shoppers.
Purpose of PERMAP:
The use of MDS for the construction of perceptual maps is well developed and
several computer programs are available. In fact, MDS was one of the earliest
uses of high-speed computers in psychology and the social sciences. The purpose
of PERMAP is to provide a particularly convenient method of producing
perceptual maps and to do so in a way that helps the researcher avoid a number
of common mistakes, as described in following sections.
Usefulness of perceptual maps:
A major advantage of MDS and perceptual maps is that they deal with
problems associated with substantiating and communicating results based on data
involving more than two dimensions. They discussed the importance of graphical
communications and the role of the eye in interpreting and distinguishing
object (factor, stimulus, characteristic) grouping.
Although experts may be able to
extract the subtle relationships represented in a matrix of numbers, this skill
is not widespread. Another important aspect of perceptual maps is that they are
forgiving of missing or imprecise data points. Whereas some analytical
techniques cannot tolerate missing elements in the input matrix, MDS results
are often unaffected. This is because it is not uncommon for there to be much
redundancy in the information given by a complete matrix of dissimilarities.
Existing perceptual mapping
difficulties: Although the theory behind making
perceptual maps is well developed, its application has been controversial There
are four major concerns that PERMAP can help alleviate. These include avoiding
local minima (i.e., configurations that are optimal with respect to small
changes in configuration but not optimal with respect to all possible changes),
proving complete convergence, minimizing the influence of outliers, and combining
multiple correlation matrices. With care, batch-operated programs can be used
in such a manner that all of these difficulties are properly addressed, but
moving to a visually interactive program renders these difficulties easier to
deal with.
PERMAP provides an interactive,
visual system for the construction of perceptual maps from multidimensional
dissimilarity data. It can treat up to 30 objects and can aggregate an
unlimited number of matrices (cases) describing the pair wise differences or
similarities among the objects. Aggregation can be accomplished using any of
three methods, and the use of weighting factors is available.
PERMAP was designed to be simple and
easy to use by a novice and to offer enough advanced features that it would be
of value to the expert. Its major improvement over existing perceptual mapping
programs is that it was designed specifically to combat certain common errors
associated with multidimensional scaling. For instance, it is particularly
effective at showing incomplete convergence, trapping by a local minima, and
outlier influence. It is also effective at revealing the importance, or lack of
importance, of the choice of the distance metric used in the objective
function. Overall, the program provides a means for the researcher to go beyond
just finding a solution to developing a feel for the suitability, stability,
and variability of the solution.
Then we started working on a new phenomenon known as Z-Scores, which was very much in continuation with our previous learnings of mean & standard deviation.
Lets know what are Z-Scores?
Sometimes we want to do more than
summarize a bunch of scores. Sometimes we want to talk about particular scores
within the bunch. We may want to tell other people about whether or not a score
is above or below average. We may want to tell other people how far away a
particular score is from average. We might also want to compare scores from
different bunches of data. We will want to know which score is better. Z-scores
can help with all of this.
They
Tell Us Important Things
Z-Scores tell us whether a
particular score is equal to the mean, below the mean or above the mean of a
bunch of scores. They can also tell us how far a particular score is away from
the mean. Is a particular score close to the mean or far away?
If
a Z-Score….
ü Has
a value of 0, it is equal to the group mean.
ü Is
positive, it is above the group mean.
ü Is
negative, it is below the group mean.
ü Is
equal to +1, it is 1 Standard Deviation above the mean.
ü Is
equal to +2, it is 2 Standard Deviations above the mean.
ü Is
equal to -1, it is 1 Standard Deviation below the mean.
ü Is
equal to -2, it is 2 Standard Deviations below the mean.
How typical a particular score is
within bunch of scores? If data are normally distributed, approximately 95% of
the data should have Z-score between -2 and +2. Z-scores that do not fall
within this range may be less typical of the data in a bunch of scores.
Individual scores from different bunch
of data. We can use Z-scores to standardize scores from different groups of
data. Then we can compare raw scores from different bunches of data.
We worked out an example related to cars having different Engine sizes, horse power & mileages & calculated in Excel. So here we will discuss how can we calculate Z-Scores using SPSS software.
I’m going to use this
example to help you understand how to enter the data. You can follow along
first and then enter your own data by using the same steps. Just change the
data points of course. Suppose you want to know how well you are doing relative
to everyone else in your class on your first test. Here are all the test scores
for your class, including yours.
·
Nidhi = 89%
·
Nikita = 75%
·
Pallavi = 50%
·
Palak = 90%
·
Nitin = 81%
·
Nitesh = 65%
·
You = 98%
·
Nishidh = 94%
·
Nihal= 84%
·
Neha = 70%
Enter all test scores into the first column of cells
Simply type in each test score and hit the enter key. Every time you hit enter, the cursor will move to the cell below the one you are currently working in
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In this example, you
can see all of the test scores in one column. For example, the number 89 in the
top cell is the test score of Nidhi. The number 70 in the bottom cell is the
test score of Neha.
Naming your
variable
It’s really important
to name variables for data that you enter. This helps you keep track of them
later on when you are trying to analyse Your current variable name will appear
at the top of the column of data.. To rename your variable, double click on the
box that says var00001.
A Define Variable box will appear
This box will let you modify some things about your variable. This includes the variable name. Look in the Variable Name box. You can see that right now, the variable name is still var00001.
Give your
variable a meaningful name
To change the name,
type another name into the box. Make sure the name is meaningful to you and
that it describes your variable. Since I am giving an example about grades
data, I will name my variable Grades.
Check out what
happens to your variable name
Success
Congrats! You have
just entered data that can be transformed into Z-Scores. But wait. Most
students think that they are finished when they name the variable. Not so.
There is one last think to do and it is so important.
Save your data
to a meaningful place with a meaningful name
Analyze
Click “Analyze,”
“Descriptive Statistics,” and then “Descriptives.”
This box will appear.
There will be two big windows in this box, one on the right and one on the
left. You should see your variable name in the box on the left. Your goal will
be to move your variable name to the Variables box on the right. To do this, click
on your variable name to highlight it. Next, click the arrow button.
Your variable name
should move to the box on the right. The next step will be for you to check the
box labeled “Save standardized values as variables.” To put a check in this
box, simply click the box with your mouse.
Click OK
When you are
finished, click the OK button and wait a few seconds for processing.
You
will see some descriptive statistics for your data set like the number of
scores (N), minimum and maximum score, the mean and the Standard Deviation.
This file will NOT contain your Z-scores. However, you may want to remember the
mean so write it down if that’s the case.
So where are the Z-scores
It’s interesting. The
Z-scores do not appear in the output file. They actually appear instead in the
data file that you had created earlier. This data file should still be open.
Click this data file to view it.
In your data
file
You will see two
columns in your data file. The first column is the one that you created. It
contains the name that you gave it and the scores that you entered. In our
example, the first column is still “grades” and still contains all of the exam
grades from students in the class. The second column is new. It is something
that was generated by SPSS when we conducted our analysis. The second column
contains Z-scores. SPSS will name the second column for you. It will give the
second column the same name as the first column with a letter ‘z’ in front of
it. In our example, the second column is named ‘zgrades’ to tell you that it
contains all of the Z-scores for the grades column.
Save the Data
file again
You data file has
changed since you conducted this analysis. So, it’s a good idea to save this
file again. Click ‘File’ and then ‘Save.’ If you do this, your file will be
saved under the same name that you chose originally.
Look in your
data file
The data that you
typed in will appear in the left most column. Your z-scores will appear in a
second column of data with the letter ‘z’ in front of its name. In our example,
the second column is named ‘zgrades’ to tell you that it contains all of the
Z-scores for the grades column.
Identifying Z-Scores
Each data point that
you entered in the column on the left will have a corresponding z-score printed
in the column just next to it. In our example, the first score that we entered
was a grade of 89. You can see this score at the top of the left most column.
The z-score for this raw score of 89 is 0.64 (when rounded). You can see the
0.63594 at the top of the second column. So each z-score will be printed right
next to each raw score.
Positive and
Negative Z-Scores
Some z-scores will be
positive whereas others will be negative. If a z-score is positive, its’
corresponding raw score is above (greater than) the mean. If a z-score is
negative, its’ corresponding raw score is below (less than) the mean.
If a Z-Score
has a Positive Value…
This means that it is
above the group mean. See all of the positive z-scores in our example? For
example, look at the top most z-score, 0.64. It is positive because it does not
have a negative sign in front of it. This z-score corresponds with the exam
score of 89%. Because the z-score is positive, we can conclude that the exam
score of 89% is above the group mean. This means that the person who scored a
89% performed better than average.
If a Z-Score
has a Negative Value…
This means that it is
below the group mean. See the three negative z-scores in our example? They are
the ones with the negative sign in front of them. For example, look at the
bottom most z-score, -0.65. This z-score corresponds with the exaam score of 70%.
Because the z-score is negative, we can conclude that the exam score of 70% is
below the group mean. This means that the person who scored a 70% performed
less than average.
Z-Scores and
Standard Deviation
The absolute value of
the z-score tells you how many standard deviations you are away from the mean.
If a z-score is equal to 0, it is on the mean. If a Z-Score is equal to +1, it
is 1 Standard Deviation above the mean. If a z-score is equal to +2, it is 2
Standard Deviations above the mean. If a z-score is equal to -1, it is 1
Standard Deviation below the mean. If a z-score is equal to -2, it is 2
Standard Deviations below the mean. In our example, your score was 98%. This
raw score had a corresponding z-score of +1.24. The value of this z-score tells
us that your raw score of 98% was 1.24 standard deviations away from the mean.
Further, if we consider the positive sign, we can see that your raw score is
1.24 standard deviations above the group mean. This means that raw score of 98%
is pretty darn good relative to the rest of the students in your class.
Z between -2
and +2
95% of scores are
going to be no more than 2 standard deviation units away from the mean. That
means that most scores will fall between z=-2 to z=+2. However, some scores
will be greater than the absolute value of 2. You can interpret these scores to
be very far from the mean
The second session of today started at 3:30 P.M. And the major agenda was to know about bubble graph & how to use the bubble graph?
Lets get to know what is a Bubble Graph?
A bubble chart is a type
of chart that displays three dimensions of data. Each entity
with its triplet (v1, v2, v3)
of associated data is plotted as a disk that expresses two of the vi values
through the disk's xy location and the third through its size.
Bubble charts can facilitate the understanding of social, economic, medical,
and other scientific relationships.
Bubble charts can be considered a variation
of the scatter plot, in which the data points are
replaced with bubbles. As the documentation forMicrosoft Office explains, "this type of chart can
be used instead of a Scatter chart if your data has three data series, each of
which contains a set of values".
According to Berman (2007), bubble charts can
"be used in project management to
compare the risk and reward among projects. In a chart each project can be
respresented by a bubble,the axis can represent the net present value and
probability of success and the size of the bubble can represent the overall
cost of the project".
Example:
This
bubble chart displays a fictitious project portfolio. Individual project
bubbles are distinguished by their colors and patterns. The chart is divided
into equal quadrants to identify relative project attractiveness. Larger
bubbles in the upper left quadrant represent the most attractive projects while
smaller bubbles in the lower right quadrant represent the least attractive
projects. Bubbles with an "X" indicate that the bubble size
represents a negative value for NPV. This chart was created using Bubble Chart
Pro™ software.
References: http://en.wikipedia.org/wiki/Bubble_chart
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