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The Effect of Reading Comprehension and Problem So (1)

2.4 Data Analysis 
Because the dependent variable of the research consisted of two groups with high and low problem solving success, it 
has a categorical structure and therefore logistic regression analysis was used for the first sub-problem. The main 
objective of logistic regression analysis is to form a regression equation to be used in estimating which group an 
individual belongs to (Çokluk, Şekercioğlu and Büyüköztürk, 2010). In other words, it is classification of individuals 
into different groups. In this study, dual logistic regression analysis was used to determine the independent variables 
group that best explains the case of being a member of certain groups stated as two-category dependent variable. Since 
the research was exploratory, forward likelihood method was used.
Because it was aimed to determine the order of importance of contribution of the strategies to classifying successful and 
unsuccessful students in the second sub-problem of the research, and also because the dependent variable is a 
categorical variable, discriminant analysis was used. Discriminant function is used to classify individuals or units, to 
test the theories about whether individuals or units can be classified upon estimations, to research the differences 
between groups, to assess the relative order of importance of independent variables in classifications by dependent 
variable and to eliminate the insignificant variables with little importance in classification (Çokluk, 2012).
3. Findings 
In this research, students with high and low problem solving success were compared in terms of reading comprehension 
(speed, reading accuracy percentage, prosody, literal comprehension and inferential comprehension) and strategies they 
used while solving a problem.
In the study, reading comprehension skills predictive of high and low problem solving success were analysed first. 
Since logistic regression analysis was conducted for the first problem, whether there was a multiple relation problem 
between the variables was checked first. The multiple relation problem between the predictive variables in the research 
were analysed in terms of eigenvalues, case indices and variance rates. Findings were presented in Table 3.
Table 3. Analysis of multiple relation problem between the predictive variables in the research in terms of eigenvalue, 
case indices and variance rates 
Dimension 
Eigenvalue 
Case
Index 
Variance Rates 
Speed 
Reading 
Accuracy 
Percentage 
(Wrr) 
Prosody 
Literal 
Comprehension 
Inferential 
Comprehension 

5.726 
1.00 
.00 
.00 
.00 
.00 
.00 

.109 
7.26 
.00 
.00 
.53 
.23 
.07 

.073 
8.86 
.02 
.00 
.31 
.58 
.07 

.065 
9.40 
.01 
.00 
.07 
.08 
.81 

.027 
14.53 
.72 
.00 
.02 
.00 
.04 

.009 
21.17 
.25 
.92 
.16 
.01 
.00 
According to Table 3, it is seen that the eigenvalues of the predictive variables look alike and there is no value bigger 
than the others among case indices. In terms of variance means, the highest variance of each predictor is seen to be 
loaded on a different eigenvalue. This allows us to see that each predictive variable explains a different dimension of the 
variable. Table 4 shows the findings about multiple relation problem in terms of standard error, tolerance and VIF 
values.


Journal of Education and Training Studies Vol. 5, No. 6; June 2017 
52 
Table 4. Analysis of multiple relation problem between the predictive variables in the research in terms of standard error, 
tolerance and VIF values 
According to Table 4, it is seen that tolerance values of all variables are above .1. According to Mertler and Vannatta 
(2005), the fact that tolerance values are above .1 shows that there is no multiple relation problem. In this context, it was 
seen that there was no multiple relation problem among the variables in terms of tolerance values. In terms of VIF 
values of each variable, it is seen that the values are below 10 and average VIF value is 1.477. According to Çokluk 
(2010), in order for VIF values not to cause multiple relation problems, it should be below 10. In this context, it was 
seen that there was no multiple relation problem in terms of VIF values either. In terms of correlation values among the 
variables, it is seen that there are medium level relations between the variables and no high level relations. This helps us 
to see there were no multiple relation problems among the variables. Since no multiple relation problems were seen 
among the variables, first -2LL value of null model (the model with no predictive variables) and the model’s iteration 
history were checked and the findings can be seen in Table 5.
Table 5. Initial Model Iteration History 
Iteration 
-2Log Likelihood (-2LL) 
Coefficients 
Constant 
Step 0 

385.194 
-.151 

385.194 
-.151 

385.194 
-.151 
According to Table 5, -2LL is 385.194 before predictive variables are added to the model. Çokluk (2010) states that for 
excellent consistence, -2LL value should be 0 and any drops in this value contributes to the model positively. It is 
expected that at further stages, as predictive variables enter the model, this value will fall. After this stage, first 
classifications obtained as a result of logistic regression were checked and findings can be seen in Table 6. 
Table 6. First classification case obtained after logistic regression 
Real/observed case 
Estimated case 
Accurate 
Classification 
Percentage 
Low Success 
High Success 
Low Success 
150 

100.0 
High Success 
129 

.0 
Total Accurate Classification Percentage 
53.76 
According to Table 6, it is seen that all the students were classified in low success category and thus accurate 
classification percentage is 53.76%. This classification was only produced for the initial model with constant term and 
only low success students were classified accurately. The fact that the number of low success students was high in the 
model caused this rate to appear high; if the students had been classified according to high problem solving success 
category, this rate would have been 46.24% (129/279). As predictive variables are added to the model, it is expected that 
accurate classification percentage will rise. At the next step, standard error about the constant term constituting the 
initial model, Wald statistics that tests significance of the variable, degree of freedom for Wald statistics, and 
exponential logistic regression coefficient Exp(ß) value representing significance level of this value and Odds rate were 
checked. Table 7 shows the findings. 
Table 7. Variables in the initial model/equation 
Step 0 
ß 
Standard Error 
Wald 
sd 

Exp(ß) 
Constant 
-.151 
.120 
1.578 

.209 
.860 
According to Table 7, it is seen that Wald value for the initial model is not significant (wald=1.578, p >.05) and that 
exponential logistic regression coefficient Exp(ß) value representing odds ratio is . 860. When predictive variables are 
added to the initial model, the changes in Wald and Odds ratios will let us see the effect of predictive variables. At the 
next step, information is provided about scores of the predictive variables not included in the initial model, their degrees 
of freedom and error chi-square values.
Predictive Variables 
ß 
Tolerance 
VIF 
Correlations 





1.Speed 
.008 
.564 
1.774 

.636
**
.346
**
.354
**
.363
**
2.Wrr
.093 
.479 
1.589 
.636
**

.528
**
.373
**
.301
**
3.Prosody 
.037 
.720 
1.389 
.346
**
.528
**

.448
**
.549
**
4. Literal Comprehension 
.536 
.831 
1.203 
.354
**
.373
**
.448
**

.519
**
5. Inferential Comprehension 
.892 
.784 
1.276 
.363
**
.301
**
.549
**
.519
**



Journal of Education and Training Studies Vol. 5, No. 6; June 2017 
53 
Table 8. Variables not included in the initial model/equation 
Step 0 
Variables
Score
sd

Speed
23.327 

.000 
Wrr 
18.715 

.000 
Prosody
9.372 

.002 
Inferential comprehension
72.181 

.000 
Literal comprehension 
39.191 

.000 
Error chi-square statistics (x
2
ßo

86.110 

.000 
According to Table 8, it is seen that error chi-square value is significant (x
2
ßo
=86.110, p<.01). The fact that this value is 
significant enables us to see that addition of one or more variables not included in the initial model to the model will 
increase predictive power of the model. Score variables are the effect score statistics of Roa and the fact that these 
values are significant means that the variables will contribute to the model (Field, 2005). The fact that all variables in 
the model are significant shows that they can contribute to the model. Meanwhile, in the stepwise method, the 
inferential comprehension variable with the highest score statistics (72.181) is the first variable to enter the model 
followed by literal comprehension variable (39.191). After this stage, in order to see the difference between the 
chi-square values of the initial and target models, Omnibus test was conducted. Findings are shown in Tablo 9.
Table 9. Omnibus test of model coefficients 
Step 
Chi-square 
Sd 


Step 
82.536 

.000 
Block
82.536 

.000 
Model 
82.536 

.000 

Step 
17.430 

.000 
Block
99.966 

.000 
Model 
99.966 

.000 
According to Table 9, it is seen that two variables entered the initial model and led to positive contribution to the model. 
In each step, increases in chi-square statistics also confirm this finding. At the next step, consistency statistics of the 
target model were checked. Table 10 shows the findings.
Table 10. Summary of the target model 
Step 
(-2LL) 
Cox&Snell R

Nagelkerke R


302.658 
.256 
.342 

285.228 
.301 
.402 
Before analysing Table 10, noting that -2LL value of the initial model was 385.194, it is seen that 2LL value decreased 
at (385.194-302.658) 82.536 rate when the first variable entered the initial model. When the second variable entered the 
model, -2LL value is seen to decrease (302.658-285.228) 17.430 more. The fact that both variables added to the initial 
model caused significant drops in -2LL value shows that these variables are consistent with the model. According to 
Field (2005), Cox&Snell R2 and Nagelkerke R2 values both show the variance amounts models explain in the 
dependent variant. According to Field, Nagelkerke R2 values appear to be higher than Cox&Snell R2 values. It is seen 
that according to Cox&Snell R2 values, inferential comprehension variable that entered the model first explained 26% 
of the total variance but when literal comprehension skill entered the analysis in the second step, both variables together 
explained 30% of the change in the variance. When this rate is analysed in terms of Nagelkerke R2 values, these three 
variables together explained 40% of the change in the variance (first step=.34, second step=.40). In the following stage, 
to assess consistency of the model as a whole, Hosmer and Lemeshow tests were conducted. Table 11 shows the 
findings 
Table 11. Results of Hosmer and Lemeshow tests 
Step 
Chi-square 
Sd 


13,131 

,041 

10,196 

,252 
According to Hosmer and Lemeshow test results in Table 11, the results are significant for the first step (p<.05), but not 
significant for the second step (p>.05). According to Çokluk (), the fact that Hosmer and Lemeshow test results are 
significant shows that model-data consistency is not acceptable and the fact that Hosmer and Lemeshow test results are 
not significant shows that model-data consistency is acceptable. In this context, it turns out that data consistency of the 
first model is not acceptable but that of the second model is acceptable. In the next stage, another indicator of model 
consistency, that is the classification table that allows comparison of the real cases of the subjects with the group in 
which they appear in the model, was checked and Table 12 shows the findings. 


Journal of Education and Training Studies Vol. 5, No. 6; June 2017 
54 
Table 12. Classification table 
Real/observed case
Expected Case 
Accurate Classifying 
Percentage 
Low Success 
High Success 
Step 1 
Low Success 
120 
30 
80.00 
High Success 
44 
85 
65.89 
Total Accurate Classifying Percentage 
73.47 
Step 2 
Low Success 
122 
28 
81.33 
High Success 
36 
93 
72.09 
Total Accurate Classifying Percentage 
77.06 
Before analysing Table 12, considering the first classifying results, it is seen that the number of students with low 
problem solving success was 150 while the number of students with high problem solving success was 129 and since 
the group with low problem solving success was taken as reference, the accurate classifying rate was (150/279) 53.76%. 
When inferential comprehension variable was added to the model, 120 out of 150 students were seen to be classified 
accurately while 30 were classified inaccurately, which made accurate classifying percentage 80%. When the case is 
considered in terms of high problem solving success, 85 students out of 129 did accurate classification and 44 did 
inaccurate classification, which made accurate classifying rate 65.89%. Total accurate classifying percentage for the 
first step is 73.47%. ın the second step, when literal comprehension variable was added to the target model together with 
inferential comprehension variable, 122 out of 150 students were seen to be classified accurately while 28 were 
classified inaccurately, which made accurate classifying percentage 81.33%. When the case is considered in terms of 
high problem solving success, 93 students out of 129 did accurate classification and 36 did inaccurate classification, 
which made accurate classifying rate 72.09%. Total accurate classifying percentage for the second step is 77.06%. In 
the next stage, the coefficient estimations for the variables in the target model were analysed and Table 13 shows the 
findings. 
Table 13. Coefficient Estimations for the Variables in the Target Model 
Step 
ß 
Standard 
error 
Wald 
sd 

Exp(ß) 

Inferential 
Comprehension 
.500 
.068 
54.292 

.000 
1.649 
Constant 
-4.209 
.566 
55.300 

.000 
.015 

Inferential 
Comprehension 
.450 
.069 
42.316 

.000 
1.568 
Basit Anlama 
.310 
.078 
15.890 

.000 
1.363 
Constant 
-5.717 
.741 
59.549 

.000 
.003 
According to Table 13, it is seen that when the first predictive variable, inferential comprehension, entered the model, 
Wald value, which hadn’t been significant in the initial model, became significant (Wald=54.292, p<.01), and when the 
second variable, literal comprehension, entered the model, Wald value sustained its significance by increasing more 
(Wald=59.549, p<.01). These findings enable us to see that both inferential comprehension and literal comprehension 
contribute to the model. According to Çokluk (2010), 1-Exp(ß).100 formula is used to determine how much low or high 
success odds of a variable is affected positively or negatively. This formula enables us to see that one unit increase in 
inferential comprehension variable caused 56.8% [(1-1.568).100] rise in low success odds while one unit increase in 
literal comprehension variable caused 36.3% [(1-1.363).100] rise in low success odds.
Table 14. The model when predictive variables are omitted from the model 
Variable 
Model LL 
Change in -2LL 
Sd 

Step 1 
Inferential 
Comprehension 
-192.597 
82.536 

.000 
Step 2 
Inferential 
Comprehension 
-171.192 
57.156 

.000 
Literal 
Comprehension 
-151.329 
17.430 

.000 
According to Table 14, when inferential comprehension variable enters the basic model which involves only the 
constant term, the change in -2LL value is 82.536 and when the second variable, literal comprehension, enters the 
change becomes 17.430. In both cases, the change in the consistency of the model is significant (p<.01). According to 
Field (2005), the fact that the change in -2LL value is significant is indicative of the contribution of the variables to the 
model and so omitting the variables that contribute to the model isn’t a good idea. In this context, it was decided to keep 
both inferential comprehension and literal comprehension variables in the model. As can be remembered, five variables 
had been added to the model and while inferential comprehension and literal comprehension were kept in the model, 
speed, reading accuracy rate and prosodic reading variables couldn’t enter the model.


Journal of Education and Training Studies Vol. 5, No. 6; June 2017 
55 
In this section, answer was sought to the second problem of the research: “What are the differences between problem 
solving strategies used by students showing high and low problem solving success?” For this purpose, discriminat 
analysis was used. For discriminat analysis, descriptive statistics of the strategies used by students showing high and 
low problem solving success were analysed first and Table 15 shows the findings 
Table 15. Group statistics 
Strategies 
Group 



Writing mathematical sentence 
Low success 
150 
4.90 
2.32 
High success 
129 
1.72 
1.54 
Looking for a pattern
Low success 
150 
.84 
.74 
High success 
129 
1.45 
.59 
Systematic listing 
Low success 
150 
.06 
.28 
High success 
129 
.79 
.88 
Estimation and control 
Low success 
150 
.37 
.72 
High success 
129 
1.87 
1.31 
Backward Studying
Low success 
150 
1.14 
.78 
High success 
129 
1.40 
.70 
Drawing figures and diagrams
Low success 
150 
.84 
1.10 
High success 
129 
1.80 
1.19 
According to Table 15, use of writing mathematical sentence is more in the group with low success (X=4.90) than the 
group with high success (X=1.72). The rate of use of “looking for a pattern” strategy is more in the group with high 
success (X=1.45) than the group with low success (X=.84). The rate of use of “systematic listing” strategy is more in 
the group with high success (X=.79) than the group with low success (X=.06). The rate of use of “estimation and 
control” strategy is more in the group with high success (X=.1.87) than the group with low success (X=.37). Similarly, 
the rate of use of “backward studying” strategy is more in the group with high success (X=1.40) than the group with 
low success (X=1.14). Finally, the rate of use of “drawing figures and diagrams” strategy is more in the group with high 
success (X=1.80) than the group with low success (X=.84). These findings show that except for writing mathematical 
sentence strategy, all the other strategies are used more in the group with high success. In the next stage, eigenvalue and 
canonical correlation degree of discriminant function were analysed and Table 16 shows the findings.
Table 16. Eigenvalues 
Function 
Eigenvalue 
Variance 
Canonical correlation 

1.612 
100.0 
.786 
According to Table 16, it is seen that eigenvalue of discriminat function is 1.612 and according to Kalaycı (2005), this 
value doesn’t have a max. limit but its min. limit is .40. Accordingly, it can be said that eigenvalue of the function is 
rather high. Çokluk (2012) states that canonical correlation value gives information about separation efficiency of 
discriminant function. In this study, canonical correlation value is .79, which helps us see that this function has a strong 
effect on separating the strategy differences of the students with high and low success. In the next stage, Wilks’ lambda 
and chi-square values, other values produced for the separation efficiency of the function, were analysed and Table 17 
shows the findings.
Table 17. Wilks’ Lambda statistics 
Function 
Wilks’
Lambda 
Chi-square 
Sd 


.383 
263.038 

.000 
According to Table 17, Wilks’ Lambda value is .383 and according to Çokluk (2012), if this value converges to 1, it 
means that separation efficiency of the function is inadequate and if the value decreases, it shows that separation 
efficiency increases. The fact that Wilks’ Lambda value was low shows that separation efficiency of the model is high. 
Another value for separation efficiency is chi-square value and this value is seen to be significant for our function [X

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