(i) How many reliable and interpretable components are there among the variables?
Two tests of assumptions to determine the suitability of factor analysis were carried out and these are: Kaiser-Meyer-Olkin‟s test of sampling adequacy and Bartlett‟s test of sphericity.
Table 1 shows the results of the two tests.
Table 4.1: Tests of Assumptions of Factor Analysis
KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .690 Bartlett's Test of Sphericity Approx. Chi-Square 2.805E4
Df 6328
Sig. .000
Kaiser-Meyer-Olkin‟s (KMO) measure of sampling adequacy shows that the value is 0.690. KMO values vary between 0 and 1. Values closer to 1 are better. A value of 0.6 is suggested as minimum loading. Thus factor analysis can be conducted based on KMO‟s test.
Bartlett‟s test of sphericity tests the null hypothesis that the correlation matrix is an identity. The decision here is that for factor analysis to be tenable, the decision is to reject the null hypothesis.
Table 4.1 shows that the value is significant and as such the hypothesis that it is an identity is rejected. Thus, factor analysis can be conducted. Taken together, these tests provide a minimum standard which should be passed before a factor analysis (or a principal components analysis) is conducted.
The correlation matrix of the analysis shows that Fifty-six (62.22%) out of the possible ninety pairs of correlation shown here are significant at.05 levels. Most of the bivariate correlations are positive with only eight (8.88%) out of the ninety having negative values
Table 4.2 shows the communalities values for all the initial components. This is the proportion of each variable's variance that can be explained by the factors (e.g., the underlying latent continua). It is also defined as the sum of squared factor loadings for the variable.
61 Table 4.2: Communality Values
items Initial Extract items Initial Extract items Initial Extract Items Initial Extract 1 1.000 .678 29 1.000 .582 57 1.000 .648 85 1.000 .616 2 1.000 .683 30 1.000 .628 58 1.000 .625 86 1.000 .614 3 1.000 .661 31 1.000 .620 59 1.000 .609 87 1.000 .637 4 1.000 .634 32 1.000 .557 60 1.000 .598 88 1.000 .616 5 1.000 .654 33 1.000 .637 61 1.000 .564 89 1.000 .586 6 1.000 .579 34 1.000 .577 62 1.000 .529 90 1.000 .605 7 1.000 .682 35 1.000 .604 63 1.000 .633 91 1.000 .580 8 1.000 .643 36 1.000 .590 64 1.000 .633 92 1.000 .577 9 1.000 .610 37 1.000 .637 65 1.000 .576 93 1.000 .592 10 1.000 .614 38 1.000 .603 66 1.000 .606 94 1.000 .604 11 1.000 .642 39 1.000 .587 67 1.000 .621 95 1.000 .625 12 1.000 .588 40 1.000 .596 68 1.000 .581 96 1.000 .596 13 1.000 .590 41 1.000 .593 69 1.000 .641 97 1.000 .576 14 1,000 .598 42 1.000 .605 70 1.000 .585 98 1.000 .638 15 1.000 .667 43 1.000 .532 71 1.000 .633 99 1.000 .538 16 1.000 .656 44 1.000 .505 72 1.000 .582 100 1.000 .533 17 1.000 .608 45 1.000 .612 73 1.000 .596 101 1.000 .641 18 1.000 .652 46 1.000 .537 74 1.000 .621 102 1.000 .635 19 1.000 .591 47 1.000 .578 75 1.000 .647 103 1.000 .550 20 1.000 .541 48 1.000 .589 76 1.000 .597 104 1.000 .612 21 1.000 .621 49 1.000 .665 77 1.000 .575 105 1.000 .620 22 1.000 .634 50 1.000 .561 78 1.000 .643 106 1.000 .580 23 1.000 .602 51 1.000 .622 79 1.000 .594 107 1.000 .580 24 1.000 .601 52 1.000 .634 80 1.000 .535 108 1.000 .551 25 1.000 .605 53 1.000 .592 81 1.000 .591 109 1.000 .524 26 1.000 .604 54 1.000 .653 82 1.000 .638 110 1.000 .579 27 1.000 .672 55 1.000 .613 83 1.000 .612 111 1.000 .587 28 1.000 .640 56 1.000 .597 84 1.000 .550 112 1.000 .583 113 1.000 .496
62
The communality estimates ranged from 0.496-0.683. For item 2 with a communality estimate of 0.683, implies that 68.3% of the item which is adequate time distribution can be predicted from data on the remaining one hundred and twelve items. This implies that from the remaining one-hundred and twelve characteristics, we could determine the incidence of adequate time distribution behaviour for teaching effectiveness scale within 68.3% of the true value on the average. For item 113 with the least communality estimate of 0.496, this implies that 49.6% of the item which is to relate course content to other field and real life situation can be predicted from data on the remaining one-hundred and twelve characteristics. By knowing teaching effectiveness scale on the remaining one-hundred and twelve characteristics, we could determine the incidence of relating course content to other field and real life situation for teaching effectiveness scale within 49.6% of the true value on the average. On the whole, the communality estimates are considered high for a large number of characteristics as sixty characteristics(53.09%) possess communality estimate of 0.600 and above with fifty-two characteristics(46.02%) possess communality estimate of 0.500 and above while only one characteristic(0.89%) possesses a communality estimate of below 0.500.
Four criteria were used in determining reliable and interpretable components and these are: Eigen values greater than 1, retaining 70% of the components accounting for the total variance, Scree plot test, and ensuring that the discrepancy of the residuals greater than 0.05 between observed and reproduced correlation is kept at the barest minimum. Forty-four components had Eigen value greater than 1 as shown in Table 4.3 and this criterion is overlooked because it lacks simplicity and more so Cattell (1966) and Kaiser‟s rule(1970) do not seems to be important when the number of variables are greater than thirty.
The second component has to do with retaining 70% of the components accounting for the total variance. In this study, 60% of the variance could only be accounted for with the forty-four components. Forty-four components are still too large as the issue of parsimony with regard to data reduction has been defeated, and as such, the forty-four components were categorized by the researcher into four components that are closely related.
63
Table 4.3.1Initial Eigen values and Percentage of Variance Explained by Each Component
Compone nt
Initial Eigen Values Extraction Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative %
1 3.614 3.198 3.198 3.614 3.198 3.198
2 2.639 2.335 5.533 2.639 2.335 5.533
3 2.343 2.073 7.607 2.343 2.073 7.607
4 2.270 2.009 9.616 2.270 2.009 9.616
5 2.157 1.909 11.525 2.157 1.909 11.525
6 2.079 1.840 13.365 2.079 1.840 13.365
7 2.006 1.775 15.141 2.006 1.775 15.141
8 1.940 1.717 16.858 1.940 1.717 16.858
9 1.875 1.660 18.517 1.875 1.660 18.517
10 1.860 1.646 20.164 1.860 1.646 20.164
11 1.806 1.598 21.761 1.806 1.598 21.761
12 1.752 1.551 23.312 1.752 1.551 23.312
13 1.735 1.535 24.847 1.735 1.535 24.847
14 1.687 1.493 26.340 1.687 1.493 26.340
15 1.631 1.444 27.784 1.631 1.444 27.784
16 1.616 1.430 29.214 1.616 1.430 29.214
17 1.578 1.397 30.611 1.578 1.397 30.611
18 1.533 1.356 31.967 1.533 1.356 31.967
19 1.501 1.328 33.295 1.501 1.328 33.295
20 1.471 1.302 34.597 1.471 1.302 34.597
21 1.462 1.293 35.891 1.462 1.293 35.891
22 1.446 1.279 37.170 1.446 1.279 37.170
23 1.411 1.248 38.418 1.411 1.248 38.418
24 1.393 1.233 39.651 1.393 1.233 39.651
25 1.373 1.215 40.867 1.373 1.215 40.867
26 1.324 1.171 42.038 1.324 1.171 42.038
27 1.300 1.151 43.189 1.300 1.151 43.189
28 1.273 1.126 44.315 1.273 1.126 44.315
29 1.260 1.115 45.430 1.260 1.115 45.430
30 1.245 1.102 46.532 1.245 1.102 46.532
31 1.221 1.081 47.612 1.221 1.081 47.612
32 1.195 1.058 48.670 1.195 1.058 48.670
33 1.186 1.050 49.720 1.186 1.050 49.720
34 1.165 1.031 50.750 1.165 1.031 50.750
35 1.147 1.015 51.765 1.147 1.015 51.765
36 1.133 1.003 52.768 1.133 1.003 52.768
37 1.118 .989 53.757 1.118 .989 53.757
38 1.088 .963 54.720 1.088 .963 54.720
39 1.083 .959 55.679 1.083 .959 55.679
40 1.072 .949 56.627 1.072 .949 56.627
41 1.052 .931 57.559 1.052 .931 57.559
42 1.038 .919 58.478 1.038 .919 58.478
43 1.023 .905 59.383 1.023 .905 59.383
44 1.007 .891 60.273 1.007 .891 60.273
64 Fig 4.1: Scree Plot of Teaching Effectiveness
65
The third criterion is to assess the Scree plot and retain all the components within the sharp descent which is shown in Fig 4.1. This criterion provides a simplistic way of providing reliable components for this study. In this diagram, the sharp descent that is realistic lies between 4 and 7 components and meaningful interpretation can be further enhanced when the residuals are looked at. The goal of data reduction is parsimony. Based on the range of components in the diagram (4-7 components), residuals are computed for that range of factors indicating the number of factors to be retained. The range of residuals for 4-7 factors computed which indicate the number of discrepancy between observed and reproduced correlation exceeding 0.05 is 1338-1371 which translates to 21.0% for the range. This implies that increasing the number of components to be retained does not increase the fit of the model.
In summary, four components that are reliable and interpretable are retained as shown in Table 4.4. Three reasons inform this decision. First, scree plot indicates that the number of reliable and interpretable components fall between 4-7. Second, the number of residuals between observed and reproduced correlations is the least when four components are considered. Third, literature indicates that most reliable and interpretable components for Teaching Effectiveness scale are four in number.
Research Question 1(ii)
If reliable components are identified, how can they be meaningfully interpreted?
In order to meaningfully interpret the components retained, the following interpretability criteria are considered. These are:
There must be at least three items with significant loadings not lower than 0.30.
Variables that load on a factor must share some conceptual meaning.
Variables that load on different factors must be seen to measure different constructs.
The rotated pattern demonstrates simple structure. There must be no case of bipolar factor that gives factor interpretation some complexity.
There must be relatively moderate to high loadings on one factor only and low loadings on the other factors.
66 Table 4.4: Rotated Component Matrixa
Components
1 2 3 4
Item81 .537 .129 -.006 -.120
Item82 .504 .144 -.059 -.061
Item65 .486 -.183 .145 .016
Item98 .-.024 .478 -.056 .178
Item100 -.011 .475 -.018 -.002
Item83 .467 .141 -.027 -.052
Item102 .463 .075 -.057 .012
Item66 .-.071 -.095 .111 .661
Item99 .-.057 .641 .000 .103
Item84 .454 .028 .009 -.037
Item97 -.031 .444 -.074 .209
Item101 .426 .082 .014 .022
Item64 -.084 .420 .119 .016
Item96 .413 -.084 -.030 .282
Item103 .451 .070 -.043 .054
Item80 .420 .116 .045 -.020
Item62 .-.079 .070 358 .124
Item113 .451 .013 -.003 .156
Item67 .-.076 -.069 .159 .442
Item78 .441 .123 -.056 .155
Item61 .209 .083 .449
-.083
item 51 .405 .055 .150 .078
item 77 .317 .100 -.005 .088
item 111 -.121 .315 .033 .126
Item13 .553 -.093 -.015 .068
Item14 -.084 .113 -.085 .487
Item34 .427 .052 .104 -.030
Item12 .409 .001 .067 .059
Item 15 -.112 .011 -.046 .343
Item 33 .321 .039 .121 .000
Item 53 .033 .028 .017 .420
Item55 .114 .295 -.056 .415
Item35 .404 .146 .116 -.100
Item57 .396 .116 .034 .003
Item56 .396 .222 -.051 .046
67 Table 4.4: Rotated Component Matrixa Continued
Item36 .392 .143 .078 -.136
Item54 .090 .038 -.025 .390
Item2 .378 -.068 .251 .016
Item32 .004 .056 .048 .368
Item58 .366 .128 -.023 .059
Item52 .330 .141 .115 .051
Item8 -.078 .323 .174 .052
Item20 .604 -.088 .104 -.008
Item5 .601 -.061 -.007 .119
Item21 .079 .565 .-.011 -.119
Item19 .010 .559 .007 -.048
Item4 -.074 .554 ..056 .071
Item18 .051 .531 .125 -.142
Item6 -.016 .497 .022 .051
Item17 461 .183 .024 -.108
Item39 .422 -.067 .030 .118
Item 2 .417 .038 .025 .052
Item 3 - 0.83 .389 .126 .022
Item50 .385 -.043 . .104 .194
Item24 -.029 .359 .006 .119
Item37 .312 .145 .070 .017
Item93 .680 .000 .005 -.039
Item94 .017 .002 .678 -073
Item92 -.031 -.019 .025 .473
Item 95 .143 .020 .495 -.066
Item 91 .050 -.023 .462 .053
Item73 -.051 .057 .415 .043
Item48 .394 .221 .254 .073
Item72 .035 .058 -.004 .388
Item45 .004 -.096 .369 ..073
Item90 .033 -.008 .347 .020
Item27 -.020 .115 .345 .034
Item74 -.035 .082 .343 .041
Item46 .340 .103 .106 ..011
Item49 .333 .032 .181 .079
68
Table 4.4.1: Component 1 : This component containing thirty - eight (38) items is appropriately tagged Classroom Interaction. This is because most of the items centred on the teaching process in which the teacher dynamically engages the students.