analysis and interpretation -...
TRANSCRIPT
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ANALYSIS AND INTERPRETATION
5.1 INTRODUCTION
After the data collection is over, the next step in the realm of research is data
processing, the primary data collected in the field. The processing involves editing,
classification and coding of data. Editing is essential to identify the errors and omissions
of the collected data and rectify them in order to achieve homogeneity, consistency and
completeness. Data should be edited before being presented as information. This action
ensures that the information provided is accurate, complete and consistent. Data editing
can be performed manually, with the assistance of computer programming or a
combination of both techniques. Goode and Hatt (1952) define that ‘coding is an
operation by which data are organized into classes, and a number or symbols are given to
each item, according to the class in which it falls’. Each answer to a particular question
must be given a distinctive code or value. After the editing and coding of the collected
data, there is a need for classification of data for easy understanding. It is the first step in
the process of analysis and interpretation of data. The classification and tabulation of data
is essential for proper and systematic arrangement and presentation of data. Stockton and
Clark (1975) defined that the process of grouping a large number of individual facts of
observation on the basis of similarity among the item is called classification. Thus the
process of good classification should have clarity, homogeneity and equality of scale,
purposefulness and accuracy.
The method of investigation presented in the previous chapter helped the
investigator generate data for the present study. The generated data were coded and
grouped for verifying the hypotheses formulated for the present study. Factor analysis
and cluster analysis were employed for validation and grouping of data. Further
correlation and ‘t’ test were used to find out the relationship between the variables and to
find if there is any difference between the groups that emerged after clustering.
The analysis and interpretation in this chapter is preceded by a brief introduction
on the Factor analysis technique, and then it is followed by the analysis of hypotheses
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related to it. Next a brief description of cluster analysis is followed by it and the analysis
of hypotheses related to it and at last the test of significance and correlation test tables
with the hypotheses are displayed.
5.2 FACTOR ANALYSIS
Factor Analysis is a collection of statistical methods used to (a) analyse patterns
in a correlation matrix, (b) reduce large numbers of variables to a smaller number of
components or factors, (c) simplify analysis of highly correlated independent variables,
(d) explore observed data for the presence of theoretical variables, and (e) test hypotheses
about theoretical variables. Factor Analysis can be classified as Exploratory or
Confirmatory on the basis of the researcher’s objective. Exploratory Factor Analysis
(EFA) is used to gain insight into the structure or underlying processes that explain a
collection of variables. The term structure describes the relationships between latent
variables and measured variables. Confirmatory Factor Analysis (CFA) is used when a
researcher has a number of well- articulated theories about the latent structure of a set of
measured variables and wishes to test how well those models fit the data.
5.2.1 History of factor Analysis
The Concept of Factor Analysis can be traced back to Charles Spearman’s (1904,
1927, 1933) research on the structure of human intellect. Spearman theorized that each
measure of human ability contains a general factor, common to all other measures of
ability, and a specific component unique to itself. According to Spearman’s theory, the
only basis for a correlation between two ability measures is their shared influence of a
common factor that he called ‘g’. The earliest Factor Analysis were focused on
confirming Spearman’s general factor model and identifying tests that correlated the
highest with g, and thereby serving as measures of general intelligence. It soon became
apparent that Spearman’s one-factor theory did not accurately describe the factor
structure of ability tests.
Thurstone (1947) developed the method of multiple-factor analysis to analyse the
tests determined by more than one kind of intelligence. Unlike Spearman’s use of Factor
Analysis, Thurstone’s multiple factor analysis challenged researchers to ascertain the
number of factors required to explain a collection of test measures and then to interpret
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the factors. Modern exploratory Factor Analysis requires researchers to deal with these
same two issues, dimensionality (number of factors) and interpretability (ascribing
meaning to factors). With modern computers, exact principal-axis solutions to Factor
Analysis problems can be obtained promptly and easily. Principal-axis factors are
extracted from a correlation or covariance matrix in decreasing order of the variance
explained. The first factor explains more variance than any other possible factor; the
second factor explains the remaining variance than any other and so on.
Ad hoc decisions about the number of factors (dimensionality) were replaced with
precise rules for the number of factors (Cattell, 1958, Cattell and Vogelman,1977;
Guttman,1954; Kaiser, 1961), and transformations of the factors were introduced to
enhance interpretability (Carroll,1953; Kaiser 1958). The Kaiser- Guttman rule, which
states that a researcher should attempt to interpret the number of factors that have
eigenvalues greater than 1, became a standard. An eigenvalue measures the amount of
variance in the variables explained by a factor. Now it has become a computer program
default in the major statistical programs like SAS and SPSS.
Cattell’s (1958) Scree Test, a visual plot of eigen values, is another popular method
of determining the dimensionality of a set of variables that is the number of factors that
can be derived from the set. The most common interpretability transformation of factor
structures is Kaiser’s (1958) varimax criterion. The varimax criterion simplifies the factor
interpretation by rotating (transforming) the principal axis solution into uncorrelated
factors with maximum variation in the factor variable correlations. The varimax criterion
simplifies the interpretation of a factor by causing a separation in the variable factor
correlations. The varimax transformation, along with other analytical rotations is guided
by Thurstone’s (1947) concept of simple structure. In uncomplicated terms, a simple
structure occurs when each variable relates to only one factor.
In the second half of the 20th century, the mathematical and statistical basis of
Factor Analysis progressed to the point where rigorous tests of significance for
dimensionality and structure were possible (Bentler and Bonnet, 1980). That technical
development led to a solid statistical basis for Confirmatory Factor Analysis.
Confirmatory Factor Analysis is used when a researcher wants to evaluate a number of
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well-articulated theories about the underlying structure for a set of variables.
The researcher specifies the number of factors and how the variables relate to the factors.
Some of the key terms used in factor analysis are described below.
5.2.2 Purpose of Factor Analysis
The general purpose of factor analytic techniques is to find a way of condensing
(summarizing) the information contained in a number of original variables into a smaller
set of new composite dimensions (factors) with a minimum loss of information; that is, to
search for and define the fundamental constructs or dimensions assumed to underlie the
original variables. The four functions Factor Analysis technique can perform are as follows-
� Identify a set of dimensions that are latent in a large set of variables; that is also
referred to as R factor analysis.
� Devise a method of combining or condensing large numbers of people into
distinctly different groups within a larger population; this is also referred to as Q
factor analysis.
� Identify appropriate variables for subsequent regression, correlation or
discriminant analysis from a much larger set of variables.
� Create an entirely new set of a smaller number of variables to partially or
completely replace the original set of variables for inclusion in subsequent
regression, correlation or discriminant analysis.
5.2.3 Some of the key terms used in Factor Analysis
5.2.3.1 Factor– Factor Analysis operates by extracting as many significant factors from
the data as possible, based on the bivariate correlations between the measures. A factor is
a dimension that consists of any number of variables. Factor Analysis involves extracting
one factor and then evaluating your data for the existence of additional factors.
The successive factors extracted in Factor Analysis are not of equal strength. Each
successive factors account for less and less variance. Typically the first two or three
factors will be the strongest that account for the most variance.
5.2.3.2 Eigen value- The strength of a factor is indicated by its eigen value. Factors with
eigen values less than 1.0 usually are not interpreted.
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5.2.3.3 Factor loading- In order to determine the dependent variables constituting a
common factor, factor loadings are computed. Each factor loading is the correlation
between a measure and the underlying factor. A positive factor loading means that a
variable positively correlated with the underlying dimension extracted, whereas a
negative loading means that a negative correlation exists. By convention, loadings are
interpreted only if they are equal to or exceed plus or minus 0.30.
5.2.3.4 Rotation of factor- After obtaining factor loadings, we need to interpret them.
The factor loadings computed initially are often difficult to interpret because they are
somewhat ambiguous. Factor rotation is used to make the factors distinct. Two types of
rotation are orthogonal and oblique rotation. In orthogonal rotation, the axes remain
perpendicular. In oblique rotation, the angles between the axes, as well as the orientation
of the axes in space, may change. Generally the orthogonal rotation is preferred over
oblique rotation because the results are easier to interpret. The most popular orthogonal
rotation method is varimax. This type of rotation maximises the variance of loadings on
each factor and simplifies factors (Tabachnick and Fidell, 2001).
5.2.3.5 Principal components and principal factors analysis
Two types of factor analysis are principal components analysis and principal
factors analysis. In principal components analysis, the diagonal of the completed
correlation matrix is filled with ones. In contrast the principal factors analysis completes
the correlation matrix by entering communalities along the diagonal. Communality is a
measure of a variable’s reliability and is fairly easy to obtain after Factor Analysis.
Various techniques have been proposed for estimating communalities. The choice
between principal components and principal factor analysis rests on the goals of the
analysis. If the goal is to reduce a large number of variables down to a smaller set and to
obtain an empirical summary of the data, then principle components analysis is most
appropriate. If the research goal is driven by empirical or theoretical predications, then
principal factor analysis is the best (Tabachnick and Fidell, 2001). In the absence of any
clear information on which technique is best, we should probably use principal
components in those situations in which you do not have any empirical or theoretical
guidance on the values of the communalities.
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5.2.3.6 Exploratory Factor Analysis versus Confirmatory Factor Analysis
Tabachnickand Fidell (2001) made distinction between exploratory Factor
Analysis and confirmatory Factor Analysis. Exploratory Factor Analysis is used when we
have a large set of variables that we want to describe in simpler terms and we have no a
priori ideas about which variables will cluster together. Exploratory Factor Analysis is
often used in the early stages of research to identify the variables that cluster together.
From such an analysis, research hypotheses can be generated and tested. Confirmatory
Factor Analysis is used in later stages of the research where the researcher can specify
how variables might relate given some underlying psychological process (Tabachnick
and Fidell, 2001).
5.2.4 Factor Analysis decision diagram
Figure (5.1) shows the general steps followed in any application of factor analysis
techniques. The starting point in Factor Analysis is the research problem, next the
calculation of the correlation matrix. The correlation matrix is chosen based on the
objectives of the problem at hand. At the next stage decision has to be taken on whether
the correlation to be done between the variables or between the respondents. Factor
Analysis when applied to a correlation matrix of the individual respondents is called ‘Q’
factor analysis and when it is applied to a correlation matrix of the variables is called ‘R’
factor analysis.
After identification of ‘Q’ factor analysis or ‘R’ factor analysis at the next level
the investigator has to decide on the Factor model to be chosen whether component
analysis or common factor analysis. The component model is used when the objective is
to summarize most of the original information in a minimum number of factors for
prediction purposes. In contrast, common factor analysis is used primarily to identify
underlying factors or dimensions not easily recognised. The flow chart depicting the
following steps in Factor Analysis is shown in figure 5.1.
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Figure (5.1) Factor Analysis decision Diagram
RESEARCH PROBLEM WHICH VARIABLES TO INCLUDE?
HOW MANY VARIABLES? HOW ARE VARIABLES MEASURED?
SAMPLE SIZE?
CORRELATION MATRIX (R VERSES Q)
FACTOR MODEL
UNROTATED FACTOR MATRIX
NUMBER OF FACTORS
COMPONENT
ANALYSIS COMMON FACTOR
ANALYSIS
EXTRACTION METHOD ORTHOGONAL?
OBLIQUE?
ROTATED FACTOR MATRIX FACTOR INTERPRETATION
FACTOR SCORES FOR SUBSEQUENT ANALYSIS:
REGRESSION DISCRIMINANT ANALYSIS
CORRELATION
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In addition to selecting the factor model the investigator has to specify how the
factors are extracted either orthogonal or oblique. In orthogonal solution the factors are
extracted in such a way that the factor axes are maintained at 90 degrees, meaning each
factor is independent of all other factors. Therefore the correlation between the factors is
arbitrarily determined to be zero. In oblique solution the extracted factors are correlated.
This is based on the needs of the research problem. At this stage the investigator is ready
to extract the initial unrotated factor matrix.
By examining the unrotated factor matrix, the investigator can explore the data
reduction possibilities for a set of variables and obtain the preliminary estimate of the
number of factors to be extracted. Final determination of the number of factors is done
after the factor matrix is rotated and the factors are interpreted. The researcher may stop with
the factor interpretation or proceed to calculate the factor scores and subsequent analysis with
other statistical techniques like correlation, discriminant analysis, regression etc.
With these basics of Factor Analysis the investigator identified cognitive
processing and self-perception of learning disabilities as variables to be included in
Factor Analysis and ‘R’ factor analysis was chosen, with principle component analysis as
Factor model. Orthogonal extraction method was employed with varimax rotation.
The varimax rotation yielded four factor solutions for cognitive processing which is
described in the following sections along with the concerned Hypothesis formulated.
5.3 ANALYSIS AND INTERPRETATIONS OF FACTORS
Hypothesis 1-There will be patterns of clustering of relationships among cognitive
processing of elementary inclusive school children.
To find the artificial dimensions of cognitive processing the data matrix of
23 × 100 was considered for factor solution. In order to arrive at the new independent
factors, the 23 × 23 correlation matrix of variables related to cognitive processing was
considered and reduced to 23× 4 factor solution, which explained 68.872 % of variance
of the original variables. It was further rotated using normal varimax rotation procedure
and the four factor emerged was considered for interpretation and description.
The rotated factor matrix explained the simplified factor structure with factor loading for
each variable. Here the signs of the loadings imply the direction of association. The size
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of each loading value indicates the degree of association of each variable with the
appropriate new independent dimension. The four new independent factors which
emerged explained 68.872% of the total variance. The share of the primary factor was
found to be higher than that of the other factors.
To find the best solution in terms of interpretability and theoretical sensibility, the
interpretability was investigated using Hatcher’s interpretability criteria (Hatcher, 1994)
which read- a given component contains at least three variables with significant loadings,
a loadings of ± 0.40 being suggested as cut off point, variables loaded on the same
component share the same conceptual meaning, variables loaded on different components
appear to measure different constructs, the rotated factor pattern demonstrates ‘simple
structure’ which means that most variables load relatively high on only one component
and low on the other components and most components have relatively high factor
loadings for some variables and low loadings for the remaining ones.
As a rule of thumb that has been frequently used by factor analyst, factor loadings
greater than ±0.30 are considered significant (n= 50 or larger), ±0.40 are considered more
important, and ±0.50 or greater are considered very significant. Thus the larger the
absolute size of the factor loading, the more significant the loadings is in interpreting the
factor matrix. Hence factor loadings with values of ±0.40 or greater have been
considered for interpretation and description. The variables are tabulated on the basis of
their absolute coefficient value in the descending order for all the 4 dimensions. The four
factor solution which emerged as a result of Factor Analysis with their loadings is
explained in the following sections.
FACTOR 1–Mental processing
The first factor which is considered as the primary factor is the most important
component among the four factors. The first factor is a general factor with highest
percentage of variance accounting to 23.798%. The variables highly loaded in this factor
are tabulated with their respective loadings in descending order.
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Table 5.1 Shows variable with significant loadings in cognitive processing for Factor 1
Variable Significant loadings
Receptive Attention part B 0.799
Matching numbers part E 0.759
Number detection part C 0.684
Planned connections part B 0.666
Matching numbers part D 0.638
Receptive Attention part A 0.618
Sentence questions 0.590
Planned connections part A 0.586
Planned codes part B 0.563
Non-verbal matrices 0.553
Figure memory 0.471
A total of 11 variables were loaded under this factor. This primary factor has high
loading on most of the variables related to general construct of cognitive processing
i.e., planning, attention, simultaneous and successive processing, relating to cognitive
processing otherwise called mental processing. Hence it is named as ‘mental processing’.
In this general factor the variables related to Planning as shown in Table 5.1 were
Matching Numbers, Planned connections and Planned Codes etc., and those related to
attention were Receptive attention, Number detection etc., and those related to successive
processing were sentence questions. Simultaneous processing included items like
Nonverbal Matrices and figure memory. The positive loadings signifies that all the variables
are positively associated with each other which implies the four dimensions namely planning,
attention, successive and simultaneous processing are positively related.
From the above Table 5.1 we can infer that the general construct of cognitive
processing has four dimensions namely Mental processing, Planning, Attention, and
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Simultaneous and Successive processing. So the present study goes along with the
theoretical description of cognitive processing by Das and his colleagues (1994b) which
involves the four tasks namely planning the task, paying attention to it and
simultaneously and successively processing the information to give the output.
FACTOR 2- Simultaneous and successive processing
The variables related to successive and simultaneous processing shows highly
significant positive loadings in the second factor. The positive loading is the result of the
positive correlation of some of the variables with the rest of the variates. This second
factor accounts for the second highest proportion of variance i.e. 20.588%. The variables
with factor loadings in descending order are described below in Table No. 5.2.
Table 5.2 Shows variable with significant loadings in cognitive processing for Factor 2
Variable Significant loadings
Word series part B 0.854
Word series part C 0.845
Sentence repetition 0.824
Word series part A 0.810
Expressive Attention 0.580
Spatial relations 0.534
Total variables loaded highly in this factor were 6. The first four variables listed
in Table 5.2 i.e., Word series Part A, Word series B and Word series C and Sentence
Repetition clearly indicates that it is related to successive processing and the last variable
spatial relations with simultaneous processing. The predominance of above variables
obviously becomes the basis for naming this component as ‘successive and simultaneous
processing’. The variable ‘Expressive Attention’ is also loaded here which indicates that
attention is related to successive and simultaneous processing.
From the above table we can conclude that Successive and Simultaneous
processing variables are positively associated with each other. From the above table it is
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seen that successive and simultaneous processing goes together as predicted in theory and
further these two should act together in an individual so as to bring out the result.
FACTOR 3- Planning
The third factor accounts for a total variance of 14.082%. There are significant
positive loadings in this factor also. Variables and its significant loadings are listed in
descending order in the Table 5.3 below.
Table 5.3 Shows variable with significant loadings in cognitive processing for Factor 3
Variable Significant loadings
Matching numbers part A 0.872
Matching numbers part B 0.861
Matching numbers part C 0.761
Matching numbers part D 0.527
Planned codes part A 0.455
Five variables were highly loaded in this factor. From the table it is evident that
all the variables i.e., Matching Numbers Part A, B, C, D and Planned Codes are related to
planning component of cognitive processing. Hence it is named as ‘planning’.
The positive loadings of the variables indicate that the variables present a positive
relationship in Factor 3. Matching Numbers Part D is highly loaded in Factor 1 (0.638)
and Planned codes part A is loaded highly in Factor 4 (0.619), but it is considered here
also to support the other variables of ‘planning’ component of cognitive processing.
It can be summarized from the above Table 5.3 that all the planning variables are
loaded in a single factor and also they are positively associated with each other. From this
we can say that children who performed well in matching number task have equally
performed in planned codes task. And planning is central to every activity. A child has to
plan his activity prior to the performance of the task and need attention, which help him
to simultaneously and successively process the information.
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FACTOR 4 - Attention
This factor which individually explains 10.404% of variance contains the
following variables with high positive loading.
Table 5.4 Shows variable with significant loadings in cognitive processing for Factor 4
Variable Significant loadings
Number detection part A 0.828
Number detection part B 0.705
Planned codes part A 0.619
Receptive Attention part A 0.410
The number of variable loaded in this factor was four. Based on the loading
pattern, this factor is called as ‘Attention’. The first two variables i.e. Number Detection
Part A and B and the last Receptive Attention listed in the Table 5.4 relates to attention
aspect of the cognitive processing. And hence it was named as ‘Attention’. The positive
loadings of the variables indicate that they have positive relationship with each other.
Receptive Attention part A is highly loaded in Factor 1 (0.618) and it is considered here
in Factor 4 to support the other variables related to attention factor. It is evident from the
table that the variable planned code is loaded here which indicates that Planning is
necessary condition for attention.
It can be inferred from the above table that the variables related to attention are
positively related and for a child to be attentive he needs to have a plan in mind.
Attention is very much necessary for a child to perform an activity and prior to it he has
to have a plan in mind and so as to process the information to put forth the result.
The multivariate approach to the concept of cognitive processing enabled the
investigator to bring out the underlying constructs of cognitive processing. This approach
helped the investigator to summarize the underlying dimensions in the structure of the
raw data – matrix. As a result it was possible to obtain four independent factors
representing the construct of cognitive processing. These four factors have been named
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accordingly taking into account the loadings of the variable in each factor. These four
independent factors have also brought out the significant relationships with most of the
chosen variables of the present study. Therefore hypothesis No.1 which describes
‘patterns of clustering of relationships in cognitive processing’ is retained.
To summarize the respondent’s cognitive processing skills falls under four
dimensions namely Mental processing, Planning, Attention, and Successive and
Simultaneous processing as in Figure 5.2. There were positive loading in all the four
independent factors, which implies that the variables are positively associated with each
other. Thus proving the theory put forth by Das and his colleagues (1994b).
Fig.5.2 Four dimensions of Cognitive processing
Hypothesis 2- There will be patterns of clustering of relationships among the self-perception
of learning disabilities of elementary inclusive school children.
PLANNING
ATTENTION
SIMULTANEOUS AND
SUCCESSIVE ROCESSING
MENTAL
PROCESSING
COGNITIVE
PROCESSING
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For establishing the artificial dimensions of self-perception of learning disabilities
after item extraction was prone to exploratory factor analysis and varimax rotation.
The rotated factor matrix explained the simplified factor structure with factor loading.
The total percentage of variance accounted was 54.7%. The share of the primary factor
was found to be higher than that of the other factors.
Factor 1- ‘Skill of Cognition’
The first factor is a general factor with highest percentage of variance accounting
to 15.07%. The significant loadings with the items are arranged in Table 5.5, in
descending order.
Table 5.5. Variables with Significant loadings in self-perception of learning disabilities
for Factor 1
Items Significant loadings
While reading I have difficulty in understanding important things 0.735
While speaking with friends I find It difficult to speak about a particular thing
0.664
I have difficulty in solving Maths word problems 0.638
I don’t get the right word to speak while speaking with friends 0.590
I have trouble in following directions that have more than one or two steps
0.553
I tend to be clumsy and unorganised 0.536
While writing copy book, I am not able to write within the four lines
0.508
I have a poor memory 0.487
I find it difficult to plan my time 0.480
A total of 9 variables (items) were loaded in this factor. This Primary factor has
highest loadings on reading, writing, arithmetic, organisation, memory etc. so it is named
as “perception of skill of cognition.” Further the loadings were all positive which implies
that there is positive association between the variables. This factor resembles the first
factor in the artificial dimensions of Cognitive processing (Hypothesis 1) i.e., ‘Mental
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processing’. Skill of Cognition involves perception and judgement on the part of the
child. For perception to take place he needs attention and planning skill and for the
judgement the simultaneous and Successive processing. Thus ‘Skill of cognition’ goes
along with mental processing in an individual.
It can be inferred from the above table that all the variables related to reading,
writing, arithmetic, memory and organisation are positively associated with each other.
And for all these aspects cognitive skills are necessary.
Factor 2- ‘Skill of Processing’
Six items were significantly loaded under factor 2, which accounted for a variance
of 14.27%. The items with their loadings are displayed in Table 5.6.
Table 5.6 Variables with Significant loadings in self-perception of learning disabilities
for Factor 2
From the Table 5.6 it is evident that the items are significant positive loadings and
they are closely associated with each other. These items i.e., ‘I am a poor speller’, ‘While
writing I don’t get ideas to put in’, ‘I am a poor reader’, ‘I am a poor at basic mathematics’, ‘I
make mistakes while reading’, ‘I can tell a story but cannot write it’, reflect the
processing skills in children. And hence it is named as “perception of skill of processing”.
Visual processing skills should be helpful when solving geometry problems that must be
solved by looking at the problem as a whole, sequential visual processing skill should be
Items Significant loadings
I am a poor speller 0.787
While writing I don’t get ideas to put in 0.737
I am a poor reader 0.654
I am a poor at basic mathematics 0.649
I make mistakes while reading 0.606
I can tell a story but cannot write it 0.558
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instrumental in reading and writing and also when solving word problems and organizing
calculations that must be solved in a sequential fashion. This factor resembles the Factor
2- ie ‘Simultaneous and successive processing’ in the artificial dimensions of cognitive
processing of elementary school children. Visual sequential as well as simultaneous
processing skills are necessary in mastering reading, writing, arithmetic etc. If these skills
are impaired children cannot write or read properly.
To summarize items in the second factor ‘the skill of processing’ are positively
associated with each other. This factor resembles the second factor in the cognitive
processing of elementary inclusive school children. Processing skills are very much
essential in mastering the three r’s i.e. reading, writing and arithmetic.
Factor 3- ‘Skill of Expression’
Under factor 3 nine items got significantly loaded. The items were arranged in
descending order of their loading values. This factor accounted for a total variance of
12.98%. The items in the Table 5.7 below show positive loadings and so they are
positively associated with each other.
Table 5.7. Variables with Significant loadings in self-perception of learning disabilities
for Factor 3
Items Significant loadings
I find it difficult to tell the alphabets in order 0.698
I find it difficult to tell the months of the year in order 0.663
It is difficult for me to write about myself 0.584
My homework is not neat 0.566
My handwriting is poor 0.534
I have difficulty in solving Maths word problems 0.505
I reverse letters when I read 0.494
While writing, I give no space between words 0.466
I learn something today but do not remember it for the next day. 0.455
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The items in Table 5.7 reflect the expressive skills in children namely oral and
written. And so it is named as “perception of skill of expression”. Expressive skills are
required to convey message to others through words, facial expressions, and body
language or in writing. Expressive language skills or the ability to express one’s thoughts,
feelings and knowledge is extremely important in the educational setting. Poorly
developed expressive language skills create a barrier to student participation and create
difficulty in assessing how much the student actually has learned. Expressive language
deficits are seen in children with autism and other learning disabilities.
The skill of expression requires prior planning by the child. So the factor three
i.e., Planning in the artificial dimensions of cognitive processing ability is related with
this factor. If a child want to read or write (express) he has to plan first which part he has
to read or write and how he has to do it either by part method or whole method and what
are the aspects to be covered in it etc. So Planning is an essential component in the ‘skill
of expression’ whether it is oral or written aspect.
To conclude all the items in Factor 3, ‘skill of expression’ has high loadings and
is positively associated with each other. This Factor is related with the Planning aspect of
cognitive processing ability in elementary school children. In school, expressive language
difficulties will impact a student’s performance both in written and spoken language.
Without good expressive language, the child will have great difficulty showing people
what he or she actually knows. A person with an expressive language issue may actually
know the answer, but not be able to put it into words. Therapy can help with this problem
using stories, games and a variety of other methods and strategies.
Factor 4- Skill of Memory
Six items were loaded under factor 4. The total variance accounted by this factor
is 12.38 %. The items with their positive loadings are displayed in Table No.5.8 below in
descending order.
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Table 5.8 Variables with Significant loadings in self-perception of learning disabilities
for Factor 4
Items Significant loading
I cannot remember the words I have read 0.723
I while reading I interchange the letters in words 0.681
My note book is not neat 0.631
I forget what I am saying right at the middle of saying it 0.512
It is hard for me to memorize things for school 0.499
I often do not write down the assignments and forget what to do 0.449
Most of the items resemble memorization skill in children and hence it is named as
perception of ‘skill of memory”. It is evident from the table that most of the items are
positively loaded which implies that there is positive association between the items.
Memorization is an important concept in learning, and so it is important in skill of
reading, writing, maths etc. This factor is related with the fourth Factor i.e., ‘Attention’ in
cognitive processing ability of the elementary school children. Only if a child is attentive
he can store important things in the memory.
It can be inferred from the above table that items in Factor 4 reflect ‘skill of
Memory’ which are positively associated with each other. And this factor is related with
the Factor 4, ‘Attention’ of the cognitive processing abilities of elementary school
children. Only if the child is attentive he can store information in the memory i.e., short
term memory or long term memory. And also the stored up information which has to be
coded and stored can be retrieved, recalled or recognised only if the child is attentive in
this process. If the child is inattentive there may be gap in the information storage so that
the correct retrieval won’t take place.
It is seen from the above discussion of the four independent factor solution that
the first factor which is a general factor items loaded were related to reading, writing,
arithmetic, organisation, memory etc. named as ‘skill of cognition’. Next three factors
were specific factors related to ‘skill of processing’, ‘skill of expression’ and ‘skill of
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135
memory’. All the items in each of the independent factors were loaded positively which
implies that there were positive association between the items. Hence there are patterns of
clustering in self-perception of learning disabilities of elementary school children. So the
hypothesis 2 is accepted and retained.
More over Factor 1,2,3 and 4 in self-perception of learning disabilities is related
with Factor 1,2,3, and 4 of cognitive processing of elementary school children i.e.,
‘mental processing’ is related with ‘skill of cognition’; ‘simultaneous and successive
processing’ with ‘skill of processing’; ‘Planning’ with ‘Skill of expression’; and ‘Attention’
with ‘skill of memory’. The theory suggested by Das and his colleagues (1994b) that is
cognitive processing has four dimensions namely Planning, Attention, Simultaneous and
Successive Processing in PASS theory of Cognitive Processing is proved here that
Planning, Attention, Simultaneous and Successive Processing are components of
cognitive processes. The four dimensions of the self-perception of learning disabilities is
depicted in figure 5.3.
The factors which emerged from cognitive processing and self-perception of
disabilities were taken for further analysis in the following sections.
Fig. 5.3. Factorial dimension of self-perception of learning disabilities
SKILL OF COGNITION
SELF-PERCEPTION OF
LEARNING DISABILITIES
SKILL OF MEMORY
SKILL OF EXPRESSION
SKILL OF PROCESSING
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5.4 CLUSTER ANALYSIS
Cluster analysis is the name of a group of multivariate techniques whose primary
purpose is to identify similar entities from the characteristics they possess. It identifies
and classifies objects or variables so that each object is very similar to others in its cluster
with respect to some predetermined selection criteria. The resulting object clusters should
then exhibit high internal within cluster homogeneity and high external between cluster
heterogeneity. Cluster analysis is a multivariate statistical technique that can be used to
group individuals or objects into clusters based on particular characteristics that they
possess (Hair et al., 1995). When clustering individuals, the ultimate goal is to arrive at
clusters of people with homogeneous characteristics who thereby exhibit small within-
cluster (internal) variation, but at the same time exhibit large between cluster (external)
variations (Aldenderfer and Blashfield, 1984; Hair et al., 1995).
The main advantage of cluster analysis is that it enables the researcher to define a
cluster variate (i.e., the characteristic variables included in the comparison) which then
determines the commonalities and differences among and between groups and leads to
natural groupings (Hair et al., 1995). Furthermore, this approach provides an opportunity
to explore structures existing in data prior to attempting to explain why they exist.
Finally, taxonomy can be developed to help describe a population. These techniques have
been variously referred to as techniques of cluster analysis, Q-analysis, typology,
grouping, clumping, classification, numerical taxonomy and unsupervised pattern
recognition. This variety of nomenclature is due to its application in the field of diverse
disciplines such as Psychology, Zoology, Biology, Botany, Sociology, Artificial
Intelligence and Information Retrieval. Although the names differ across disciplines, they
all have a common dimension: classification according to natural relationships.
Cluster Analysis (CA) is a technique which seeks to separate data into constituent
groups. This technique is used for grouping of objects or individuals under investigation.
The objects which are subjected to cluster analysis are termed ‘entity’ or ‘ individual’.
The measurements taken on each entity are generally referred to as variables, characters
or attributes. The result of a Cluster Analysis will be number of groups, clusters, types or
classes.
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5.4.1 Objectives of Cluster Analysis
The goals of various users of clustering technique are frequently dissimilar.
Ball (1971) lists seven possible uses of clustering technique- Finding a true typology,
Model fitting, Prediction based on groups, Hypotheses testing, Data exploration,
Hypothesis generating and Data reduction. Cluster Analysis can be used to perform data
reduction procedure objectively by reducing the information from an entire population or
to set information about specific smaller subgroups. Cluster Analysis may be used to
generate hypotheses concerning the nature of the data and useful in shedding light on
previously made hypotheses. In some investigations Cluster Analysis methods may be
used to produce groups which form the basis of classification scheme useful in later
studies for predictive purposes of some kind. In general the technique of Cluster Analysis
which is a useful tool for data analysis can be used to search for natural groupings in the
data, to simplify the description of a large set of multivariate data, to generate hypotheses
to be tested on future samples and to verify the previously stated hypotheses.
5.4.2 Types of Cluster Analysis technique
The technique for Cluster Analysis seeks to separate a set of data into groups or
clusters. Cluster Analysis technique may be classified into types as follows-
• Hierarchical techniques- In Hierarchical techniques the classes themselves are
classified into groups, the process being repeated at different levels to form a tree.
• Optimization-partitioning techniques- In this technique the clusters are formed by
optimization of ‘clustering criterion’. The classes are mutually exclusive, thus
forming a partition of the set of entities.
• Density or mode - seeking techniques- In this the clusters are formed by searching
for regions containing a relatively dense concentration of entities.
• Clumping techniques- In this technique the classes or clumps can overlap.
These types are not necessarily mutually exclusive, and several clustering
techniques could be placed in more than one category. For the present study the
investigator has employed the hierarchical clustering technique and so it is presented in
detail in the following sections.
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5.4.3 Hierarchical clustering techniques
Hierarchical clustering techniques may be subdivided into agglomerative methods
which proceed by a series of successive fusions of the N entities into groups, and divisive
methods which partition the set of N entities successively into finer partitions. The results
of both agglomerative and divisive techniques may be presented in the form of a
dendrogram, which is a two-dimensional diagram illustrating the fusions or partitions
which have been made at each successive level. Since all agglomerative hierarchical
techniques ultimately reduce the data to a single cluster containing all the entities, and the
divisive technique will finally split the entire set of data into N groups each containing a
single entity, the investigator must decide at what stage in the analysis he wishes to stop
which may be based on some criteria.
5.4.3.1 Agglomerative methods-The basic procedure with all these methods is similar.
They begin with the computation of a similarity or distance matrix between the entities.
The end product of the methods is a dendrogram showing the successive fusions of
individuals, which culminates at the stage where all the individuals are in one group.
For this reason agglomerative procedures are sometimes referred to as build-up methods.
At any particular stage the methods fuse individuals or groups of individuals which are
closest (or most similar). Differences between methods arise because of the different
ways of defining distance (or similarity) between an individual and a group containing
several individuals or between two groups of individuals. Seven popular agglomerative
procedures used to develop clusters are Single linkage, Complete linkage, Unweighted
pair-group average, Weighted pair-group average, Unweighted pair-group centroid
method, Weighted pair-group centroid (median) and Ward’s method.
Single linkage (nearest neighbour) method - In this method the distance between two
clusters is determined by the distance of the two closest objects (nearest neighbours) in
the different clusters. This rule will, in a sense, string objects together to form clusters,
and the resulting clusters tend to represent long "chains."
Complete linkage (furthest neighbour) method - In this method, the distances between
clusters are determined by the greatest distance between any two objects in the different
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139
clusters (i.e., by the "furthest neighbours"). This method usually performs quite well in
cases when the objects actually form naturally distinct "clumps". This method is
inappropriate, if the clusters tend to be somehow elongated or of a "chain" type nature.
Unweighted pair-group average method - In this method, the distance between two
clusters is calculated as the average distance between all pairs of objects in the two
different clusters. This method is also very efficient when the objects form natural
distinct "clumps," however, it performs equally well with elongated, "chain" type
clusters. Sneath and Sokal (1973) introduced the abbreviation UPGMA to refer to this
method as unweighted pair-group method using arithmetic averages.
Weighted pair-group average method- This method is identical to the unweighted
pair-group average method, except that in the computations, the size of the respective
clusters (i.e., the number of objects contained in them) is used as a weight. Thus, this
method (rather than the previous method) should be used when the cluster sizes are
suspected to be greatly uneven. Sneath and Sokal (1973) introduced the abbreviation
WPGMA to refer to this method as weighted pair-group method using arithmetic
averages.
Unweighted pair-group centroid method- The centroid of a cluster is the average point
in the multi-dimensional space defined by the dimensions. In a sense, it is the center of
gravity for the respective cluster. In this method, the distance between two clusters is
determined as the difference between centroids. Sneath and Sokal (1973) used the
abbreviation UPGMC to refer to this method as unweighted pair-group method using the
centroid average.
Weighted pair-group centroid (median) method - This method is identical to the
previous one, except that weighting is introduced into the computations to take into
consideration differences in cluster sizes (i.e., the number of objects contained in them).
Thus, when there are (or we suspect there to be) considerable differences in cluster sizes,
this method is preferable to the previous one. Sneath and Sokal (1973) use the
abbreviation WPGMC to refer to this method as weighted pair-group method using the
centroid average.
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Ward's method- This method is distinct from all other methods because it uses an
analysis of variance approach to evaluate the distances between clusters. In short, this
method attempts to minimize the Sum of Squares (SS) of any two (hypothetical) clusters
that can be formed at each step. In general, this method is regarded as very efficient;
however, it tends to create clusters of small size.
5.4.4 Steps in cluster analysis
The objective of cluster analysis is to group observations into clusters such that
each cluster is a homogenous as possible with respect to the clustering variables.
The various steps in cluster analysis are-
• Select a measure of similarity
• Decision is to be made on the type of clustering technique to be used
• Type of clustering method for the selected technique is selected
• Decision regarding the number of clusters
• Cluster solution is interpreted
No generalisation about cluster analysis is possible as vast number of clustering
methods have been developed in several different fields with different definitions of
clusters and similarities. There are many kinds of clusters namely-
• Disjoint cluster where every object appears in single cluster
• Hierarchical clusters where one cluster can be completely contained in another
cluster, but no other kind of overlap is permitted.
• Overlapping clusters
• Fuzzy clusters, defined by a probability of membership of each object in one
cluster.
With these basics of cluster analysis in mind the investigator moved on with the
classification of elementary inclusive school children based on their cognitive processing
and self-perception of learning disabilities using the hierarchical clustering technique in
line with the hypotheses formulated in the methodology of investigation.
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5.5 ANALYSIS AND INTERPRETATION OF CLUSTERS
Hypothesis 3- Different groups based on the cognitive processing will emerge from the
elementary inclusive school children.
To classify the 100 subjects based on their cognitive processing skills the multivariate
statistical technique cluster analysis was employed. The descriptive typologies of the 100
subjects were obtained through computing the similarity coefficients among the subjects
considered for the study. This analytical procedure was carried out based on hierarchical
clustering agglomerative method to obtain homogeneous classifications based on cognitive
processing in elementary inclusive school children with as many groups as possible.
This procedure resulted in identifying exclusive and mutually exhaustive groups or
typologies. Further as a result of this procedure, 100 subjects has been clustered in terms
of the taxonomic distances as computed between the pairs which are illustrated in the
form of a linkage tree called Dendogram figure 5.4.
The different groups of the subjects were arrived at by drawing cut-off lines
across the dendogram. This procedure has been adopted because of the fact that there are
no accepted standards or norms of the taxonomic distance values that could be considered
as characteristic indicator to establish taxonomic category or the cluster status. In the
present analysis one cut-off line was drawn across the linkage tree figure 5.4 at the mean
of the taxonomic distance. There is no analytical solution which exists for identifying the
number of distinct groups or classes. The cut-off line drawn at the mean yielded two
major groups or clusters of the subjects. Table 5.9 shows the number of subjects clustered
in each of the groups and their mean values of cognitive processing. Further two isolates
were also present in the group. These isolates were considered for purpose of
interpretation by including in the nearest and approximate group. The different groups
then evolved are considered for further interpretation. The two major groups evolved
contained within themselves a few homogenous subgroups.
Analysis of group 1
From the Figure 5.4 and Table 5.9 it is evident that the groups and subgroups
have several unique features. When we consider the cognitive processing at the group1
level it is comprised of 52 subjects arranged themselves into distinct subgroups of
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142
different sizes based on taxonomic distances. Here there were two sub groups viz.,
subgroup one with 36 subjects and subgroup 2 with 16 subjects. However considering the
mean scores of cognitive processing of these subgroups of group 1 reveals that there is
negligible difference in mean scores which emphasises the homogeneity of the sub
groups with respect to the personal variables such as age, grade, mother’s education and
occupation, father’s education and occupation etc. But when we consider the mean scores
of various subjects we find that there is large variation between the subgroups, within
group1.Group 1 is considered as high cognitive processing group as mean scores of the
subjects in this group is high compared to the group 2.
Analysis of group 2
At the group 2 level as shown in figure there were 48 subjects. Based on the
difference of the taxonomic distances it was further divided (next level) into two major
subgroups with 18 subjects in the first subgroup and 30 subjects in the second. From the
table 5.9 of mean scores of two sub groups, it is evident that there is no much variation in
the mean scores of personal variables, but there is significant variation in the mean scores
of the achievement scores. The 18 subjects in the first major subgroups at next level were
divided into two sub groups with 8 subjects in the first subgroup and 10 subjects in the
second. The 30 subjects in the second subgroup at the next level were divided into three
subgroups with 14 subjects in the first and second subgroups respectively and two in the
third. These two in the third subgroups were considered as isolates. The subgroups also
possesses the same characteristics as found in the first group namely the mean scores of
cognitive processing which have not shown much difference between the subgroups of
group 2. Hence it was considered that these subgroups are homogeneous within the
group2.
The above description of classification of subjects based on their cognitive
processing resulted in identifying 2 distinct groups of cognitive processing in elementary
inclusive school children, thus confirming the hypothesis 2 i.e., different groups based on
cognitive processing will emerge from the elementary inclusive school children.
Therefore hypothesis 2 is retained.
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Table 5.9 Cognitive processing Taxonomic group means
Sl. No
Groups Group 1 Group 2
Subgroups Subgroup 1 Subgroup 2 Subgroup 1 Subgroup 2
Description Mean Scores
Std. Dev.
Mean Scores
Std. Dev.
Mean Scores
Std. Dev.
Mean Scores
Std. Dev.
1 Age 11.22 0.832 12.06 1.29 11.78 1.11 11.63 1.19
2 Grade 6.44 0.504 6.75 0.447 6.61 0.502 6.33 0.479
3 Mother’s education
1.28 0.513 1.06 0.250 1.06 0.236 1.00 0.00
4 Father’s education
1.25 0.500 1.00 0.00 1.28 0.669 1.03 0.183
5 Mother’s occupation
2.97 1.341 3.69 0.602 3.61 0.778 3.70 0.466
6 Father’s occupation
2.50 0.811 2.94 0.250 2.61 0.778 2.97 0.183
7 Science marks (%)
81.50 11.60 63.40 23.28 58.74 22.80 36.73 16.31
8 Maths (%) 80.45 13.38 60.24 23.88 59.17 25.81 38.42 18.09
9 Malayalam (%) 79.93 15.57 66.61 23.08 63.45 21.83 36.87 17.40
10 Social science (%)
80.52 14.33 59.22 23.40 63.14 20.89 36.86 16.43
11 English (%) 82.46 10.60 63.60 23.20 58.25 23.91 35.91 17.58
12 Hindi (%) 75.83 19.06 56.95 27.02 53.53 24.46 35.89 16.83
13 Avg. (%) 80.13 12.15 61.67 22.50 59.38 21.77 35.79 15.91
14 Total no. of subjects
36 16 18 30
From the figure 5.4 it is evident that the subgroup of group1 has the following
subjects of the elementary inclusive school children representing each of the sub groups.
A thorough introspection of the data set of the subjects of cognitive processing subgroups
clearly indicates the homogeneous characteristics of the cognitive processing within each
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144
of the subgroups or sub clusters. Further the analysis of mean scores of the variables of
the subgroups with the major group I and II Table No 5.9 clearly reveals there is
homogeneity.
Finally from Table 5.9 and Figure 5.4 of the subgroups and major groups it is
clear that the elementary inclusive school children have grouped themselves based on
their similarity of cognitive processing characteristics. They were grouped themselves
due to their individual characteristics. In this study the 100 elementary inclusive school
children at the highest level has emerged as a single homogeneous group. This implies
that they are related to each other in planning, attention or simultaneous and successive
processing dimensions of cognitive processing.
Based on the above typological description which resulted in two homogeneous
groups of elementary inclusive school children i.e., the low cognitive processing group and
the high cognitive processing groups, which confirms hypothesis 2. Hence hypothesis 2 is
retained i.e., different groups emerged from the elementary inclusive school children
based on their cognitive processing skill/abilities. Also two isolates were also spotted out
who are very weak in their processing skills but they join the subgroup in certain unique
characteristics.
Hypothesis 4- Different groups based on the self-perception of learning disabilities will
emerge from the elementary inclusive school children.
In order to find out the typologies of elementary inclusive school children on self-
perception of learning disabilities, Taxonomic procedure cluster analysis (CA) was
carried out. Based on the 32 items of self-perception of learning disabilities the subjects
were found to cluster themselves into groups. The result obtained through Cluster
Analysis is presented in the form of a linkage tree Figure 5.5.
Cut of lines were drawn at the mean of the taxonomic distances to get clear and
distinct groups. By this two major groups were formed at the first level. Different groups
and subgroups emerged from the subjects are shown in the linkage tree Figure 5.5 and in
the Table 5.10. The 100 subjects cluster themselves at the first level, to form two major
groups, with 50 subjects in the first major group and the other 50 in the second. The first
major group (50 subjects) cluster into subgroups at the second level, with 36 subjects in
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145
the first subgroup and 14 subjects in the second. At the third level the subgroups with 36
subjects are clustered into 3 more subgroups with 24 subjects in the first, 3 subjects in the
second and 9 subjects in the third. The second major groups (50 subjects) are clustered into 3
subgroups at the second level with 14 subjects in the first, 17 subjects in the second and 19
subjects in the third subgroup. At the third level the 14 subjects group clustered into 2
subgroups with 6 and 8 subjects in each, the 17 subjects groups clustered into 4 subgroups
with 2 subjects in the first, 4 subjects in the second, 7 in the third and 4 in the fourth.
Table 5.10 Showing Taxonomic group means based on self-perception of learning
disabilities
Sl. No.
Groups Group 1 Group 2
Subgroups Subgroup1 Subgroup2 Subgroup1 Subgroup2
Description Mean scores
Std. Dev.
Mean scores
Std. Dev.
Mean scores
Std. Dev.
Mean scores
Std. Dev.
1 Age 11.28 0.944 11.86 1.10 11.50 1.22 11.81 1.17
2 Grade 6.44 0.504 6.64 0.497 6.36 0.497 6.53 0.506
3 Mother’s education
1.28 0.513 1.00 0.00 1.00 0.00 1.06 0.232
4 Father’s education
1.33 0.586 1.00 0.00 1.00 0.00 1.08 0.368
5 Mother’s occupation
3.00 1.37 3.71 0.726 3.86 0.363 3.56 0.557
6 Father’s occupation
2.42 0.874 2.93 0.267 2.93 0.267 2.89 0.398
7 Science marks (%)
82.12 10.61 67.36 20.87 31.77 9.72 48.99 22.53
8 Maths (%) 79.61 15.30 67.16 22.44 32.83 10.22 50.33 24.09
9 Malayalam (%)
82.74 10.39 66.31 25.56 32.44 10.70 50.84 22.70
10 Social science (%)
80.88 11.51 65.85 23.23 31.22 10.72 50.50 22.61
11 English (%) 80.05 14.17 71.48 21.77 32.07 11.49 49.46 24.12
12 Hindi (%) 75.63 18.56 63.26 24.16 31.43 9.62 45.37 24.20
13 Avg. (%) 80.18 11.11 66.91 21.56 31.97 8.96 49.25 22.07
14 Total no. of subjects
36 14 14 36
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At level three the 19 subject group is clustered into 2 subgroups with 6 subjects
and 13 subjects in each group. At the fourth level the 6 subjects subgroups are clustered
into 2 subgroups with 3 subjects in each and the 13 subject’s subgroups formed 3
subgroups with 5, 3, and 5 subjects in each. Finally from the Table 5.10 and Figure 5.5,
it is clear that based on their similarities and differences over the 32 items of
self-perception of disabilities after applying the classification procedure individually are
found to form groups of different sizes of elementary inclusive school children. Two major
groups emerged with two isolates seen in the second group, which joins with the others to
form a subgroup. The presence of isolates indicates that they have characteristics
different from the subgroups. But in certain unique characteristics they join with the other
subgroup.
Analysing the data set of the subjects who fall between different levels and
different subgroups they are mostly homogeneous in their raw scores on self-perception
of disabilities. Also it is seen that each level differs in their raw scores but differences are
not much, because of small distance between the subjects. Also from the table of mean
scores it is seen that there is not much variation in the mean scores of the personal
variables such as age, grade, mother’s education and occupation, father’s education and
occupation etc., taken for the study but a large variation is seen between the achievement
scores of the subgroups and the groups formed after cluster analysis. This is further
proved in the last two hypotheses.
To conclude from the above analysis pertaining to the data of elementary school
children yielded homogenous groups based on self-perception of learning disabilities.
Hence the Hypothesis 4 is accepted and retained i.e., different groups will emerge from
the elementary inclusive school children based on self-perception of learning disabilities.
5.6 ANALYSIS AND INTERPRETATIONS OF RELATIONSHIPS
Hypothesis 5: There will be significant relationship between achievement and the factors
that emerged from cognitive processing among the elementary inclusive school children
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Table 5.11Correlation Coefficients of achievement and cognitive processing factors
Achievement
Factors
Science Maths Malayalam Social science English Hindi
F1 (mental processing)
0.782** 0.730** 0.713** 0.743** 0.770** 0.647**
F2 (simultaneous and successive Processing)
0.712** 0.644** 0.676** 0.686** 0.687** 0.598**
F3 (Planning )
0.513** 0.464** 0.458** 0.481** 0.476** 0.381**
F4 (Attention)
0.515** 0.495** 0.488** 0.518** 0.503** 0.417**
** Correlation significant at the 0.01 level (2tailed)
Table 5.11 above shows the correlation coefficients of cognitive processing i.e.,
planning, attention and simultaneous and successive processing and achievement scores
in the subjects Science, Maths, Malayalam, Social science, English and Hindi of the
elementary inclusive school children. It is evident from the table that the correlation
coefficients are high and positive and also significant at 0.01 level (2 tailed), which
implies that as planning, attention, simultaneous and successive processing of an
individual increases the achievement increases. This shows that if the cognitive
processing of a child is high, his achievement also will be high.
It can be summarized from the above table that there is a positive relationship
between cognitive processing and achievement. Hence the Hypothesis that there will be
significant relationship between the achievement and cognitive processing factors is
retained. Further if a child’s cognitive processing skill is high he will have high planning,
attention, and simultaneously and successively process the information. He will be able to
perform well in the academics.
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Hypothesis No.6: There will be significant relationship between achievement and factors
of self-perception of learning disabilities among the elementary inclusive school children.
Table 5.12 Correlation coefficients of achievement and self-perception of learning disabilities
Achievement
Factors
Science Maths Malayalam Social
Science English Hindi
F1 (skill of cognition)
-0.635** -0.553** -0.637** -0.568** -0.559** -0.495**
F2 (skill of processing )
-0.781** -0.748** -0.729** -0.738** -0.754** -0.668**
F3 (skill of expression)
-0.639**
-0.566** -0.655** -0.605** -0.599** -0.532**
F4 (skill of memory)
-0.653** -0.584** -0.633** -0.618** -0.617** -0.542**
** Correlation significant at the 0.01 level (2tailed)
Table 5.12 displays the correlation coefficients of achievement and self-perception
of learning disabilities (factors) among the elementary school children. There is
significant high negative correlation between self-perception of learning disabilities
factors i.e., skill of cognition, skill of processing, skill of expression and skill of memory
and the achievement scores in the subjects Science, Maths, Malayalam, Social science,
English and Hindi among the elementary inclusive school children. And it is significant
at 0.01 level of significance (2 tailed). This implies that as self-perception of learning
disabilities increases the achievement decreases and vice versa.
It can be inferred from the above table that the high negative correlation signifies
that as perception of learning disabilities in the skill of cognition, processing, expression
and memory decreases their achievement increases and vice versa which implies that
students who perceived their learning disabilities in reading, writing, arithmetic etc. as
low are good performers in academic achievement rather than the high perceivers.
In general, students who experience disabilities in reading, writing, arithmetic etc., have
low perception on learning and hence are low achievers.
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5.7 ANALYSIS AND INTERPRETATIONS OF MEAN DIFFERENCE S (t test)
Hypothesis 7: There will be significant mean score difference in achievement between
the high and low groups in factors that emerged in cognitive processing (CP) among the
elementary inclusive school children.
Table 5.13 Mean scores difference in achievement between high and low groups of
factors in Cognitive Processing
Factors Groups N df Mean SD ‘t’ value Level of
significance (0.01 level)
Mental processing (F1)
low 51 98
43.70 19.63 9.80 significant
high 49 77.86 14.79
Successive and simultaneous processing (F2)
low 50 98
45.06 21.50 8.10 significant
high 50 75.82 16.06
Planning (F3) low 52 98
48.13 21.49 6.16 significant
high 48 73.78 20.08
Attention (F4) low 51 98
48.97 24.44 5.45 significant
high 49 72.38 17.91
The above Table 5.13 presents the mean; Standard Deviation in achievement
between high and low groups of factors that emerged from cognitive processing in
elementary inclusive school children. It is evident from the table that for all the factors
i.e., mental processing, simultaneous and successive processing, planning and attention
the mean scores of the high group i.e., 77.86, 75.82, 73.78 and 72.38 is higher than the
low group i.e., 43.70, 45.06, 48.13 and 48.97. Also there is not much variation in the
Standard Deviation’s. It is clear from this that the high group on the factors of cognitive
processing has high academic achievement rather than the low group. The ‘t’ values for
factor 1, factor 2, factor 3 and factor 4 are 9.80, 8.10, 6.16 and 5.45 respectively are
found to be higher than the table value at 0.01 level of significance. And so they are
significant at 0.01 level of significance.
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Considering the mean scores it is seen that for all the factors the group with high
cognitive processing is better in achievement than the low cognitive processing groups.
This shows that cognitive processing has influence on achievement. Hence Hypothesis 7
is accepted i.e., there will be significant difference in achievement between the high and
low groups on cognitive processing.
This implies that the two groups which differ in their cognitive processing also
differ in their achievement scores. It can be inferred from the above table that the two
groups on cognitive processing differ significantly in their achievement scores; also high
cognitive processing group has higher achievement than the low cognitive processing
group. In general we can say that if cognitive processing skill/abilities of an individual
increases his academic achievement also increases.
Hypothesis 8: There will be significant mean score difference in achievement between
the high and low groups in factors that emerged from self-perception of learning
disabilities among the elementary inclusive school children.
Table 5.14 Mean scores difference in achievement between high and low groups in
factors that emerged from self-perception of learning disabilities
Factors Groups N df Mean SD ‘t’ value Level of
significance 0.01 level
Skill of cognition (F1)
low 44 98
76.60 15.50 7.24 significant
high 56 47.74 22.59
Skill of processing (F2)
low 49
98
77.62 15.10
9.54 significant
high 51 43.93 19.80
Skill of expression (F3)
low 51 98
71.68 18.51 5.31 significant
high 49 48.74 24.43
Skill of memory(F4)
low 55 98
73.49 17.96 7.31 significant
high 45 44.49 21.69
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The above table 5.14 describes the mean; SD of achievement scores of high and
low groups in factors that emerged from self-perception of learning disabilities in
elementary inclusive school children. It is seen from the above table that for all the
factors i.e., ‘skill of cognition’, ‘skill of processing’, ‘skill of expression’ and ‘skill of
memory’ the mean scores of the low group are 76.60, 77.62, 71.68 and 73.49 respectively
and found to be higher than the high group i.e., 47.74, 43.93, 48.74, and 44.49
respectively. Also there is no much variation in SD. The ‘t’ values for factor 1, factor 2,
factor 3 and factor 4 are 7.24, 9.54, 5.31, and 7.31 respectively are found to be significant
at 0.01 level of significance. It is clear from this that the high group on self-perception of
disabilities has low academic achievement scores rather than the low group. The ‘t’ value
is found to be higher than the table value for all factors and so it is significant at 0.01
level of significance. So the self-perception of learning disabilities has influence on the
achievement.
Further considering the mean scores it is evident that for all the factors the group
with low perception of learning disabilities have high achievement than the group with
high perception. This shows that self-perception influences achievement. This implies
that the two groups which differ in their self-perception of learning disabilities also differ
in their achievement scores.
It can be inferred from the above table that the two groups on self-perception of
learning disabilities differ significantly in their achievement scores. Hence hypothesis 8 is
accepted i.e., there will be significant mean score difference in achievement scores
between the high and low groups in factors that emerged from self-perception of learning
disabilities among the elementary inclusive school children. Also high perception group
has lower achievement score than the low perception group. It can be generalised from
the above that if self-perception of learning disabilities of an individual increases his
academic achievement decreases and vice versa.
Hypothesis 9: There will be significant mean score difference in achievement scores
between the high and low groups based on cluster in cognitive processing (CP) among
the elementary inclusive school children.
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Table 5.15 Mean scores difference in achievement between high and low groups of CP
Cognitive processing Groups (based on cluster)
N Mean SD Df ‘t’ value Level of significance
Low 48 45.26 21.20 98 7.437 Significant at 0.01 level (2 tailed)
High 52 74.45 18.00
The above table displays the mean; SD of achievement scores of high and low
groups of cognitive processing in elementary inclusive school children. The mean scores
of the high group on cognitive processing is found to be higher (M1=74.45) than the low
group (M2=45.26) .It is clear from this that the high group on cognitive processing has
high academic achievement scores rather than the low group. The ‘t’ value is found to be
higher than the table value and so it is significant at 0.01 level of significance. So the
cognitive processing has influence on the achievement score. Hence hypothesis 9 is
accepted and retained.
This implies that the two groups which differ in their cognitive processing also
differ in their achievement scores. It can be inferred from the above table that the two
groups on cognitive processing differ significantly in their achievement scores. Also high
cognitive processing group has higher achievement score than the low cognitive
processing group. In general we can say that if cognitive processing skill/abilities of an
individual increases his academic achievement also increases.
Hypothesis 10: There will be significant mean score difference in achievement scores
between the high and low groups (based on cluster) in self-perception of learning
disabilities among the elementary inclusive school children.
Table 5.16 Mean scores difference in achievement between high and low groups of
self perception of learning disabilities
Self-perception of disabilities
(learning) Groups (based on cluster)
N Mean SD Df ‘t’ value Level of significance
Low 50 76.47 15.74 98 8.701 Significant at 0.01 level (2 tailed)
High 50 44.41 20.75
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153
The above table displays the mean; SD of achievement scores of high and low
groups of self-perception of learning disabilities in elementary inclusive school children.
The mean scores of the high group is found to be lower (M1=44.41) than the low group
(M2=76.47). It is clear from this that the high group on self-perception of learning
disabilities has low academic achievement scores rather than the low group. The ‘t’ value
is found to be higher than the table value and so it is significant at 0.01 level of
significance. So the self-perception of learning disabilities has influence on the
achievement score. Hence hypothesis 10 is accepted and retained.
This implies that the two groups which differ in their self-perception of learning
disabilities also differ in their achievement scores. It can be inferred from the above table
that the two groups on self-perception of learning disabilities differ significantly in their
achievement scores. Also high perception group has lower achievement score than the
low perception group. In general we can say that if self-perception of learning disabilities
of an individual increases his academic achievement decreases and vice versa.
5.8 SUMMARY
The objectives and hypotheses stated in the methodology were analysed and
tested in this chapter based on the data generated after administering the tools. Four
factors emerged from each cognitive processing and self-perception of learning
disabilities. These factors scores were taken for further analysis i.e., correlation and ‘t’
test. With the cluster analysis two groups emerged each from cognitive processing and
self-perception of learning disabilities. The two groups were the high ability group on
cognitive processing and the low ability group. In self-perception of learning disabilities
using Cluster Analysis yielded the high perceivers on learning disabilities and the low
perceivers. The ten hypotheses formulated based on the study objectives were accepted
and retained.
This chapter puts forth light on the aspect that the PASS theory of cognitive
processing as stated by Das and his colleagues that PASS theory of intelligence is a
viable method in assessing the cognitive functions in children and Cognitive processing
has four dimensions namely - planning, attention, simultaneous and successive
processing. In addition to that this study yields one primary factor which includes all the
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154
three i.e. planning, attention, simultaneous and successive processing namely ‘mental
processing’. And these dimensions are highly correlated with each other i.e., as one
aspect increases other also increases and vice versa.
It is seen that as cognitive processing of an individual increases the achievement
also increases i.e., there is a positive relationship between the two. Considering the
self-perception of learning disabilities it is evident that it has a high negative correlation
with achievement, which implies that as self-perception of learning disabilities of an
individual increases his achievement decreases and vice versa. Also two groups emerged
from cognitive processing of elementary school children –the high and the low, which
were subjected to further analysis to find out if there exists any difference between the
groups. There existed a natural grouping based on self-perception of learning disabilities
i.e., the low perceived group and the high. Based on the grouping there existed a
difference in their achievement scores and self-perception.
It can be concluded from the analysis and interpretations that cognitive processing
and self-perception are related with achievement and there is significant difference in
achievement between the two groups i.e. high and low on cognitive processing and self-
perception of learning disabilities.
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