Correlation and simple linear regression methods assess the degree of strength, direction of association, and a linear summary of relationship existing between two variables, or observational units (Berg, 2004). In an effort to expose the descriptive analysis, correlational patterns resulting from the dataset DEL618_DHS618m1.sav, the writer/researcher hopes to examine the associative factors in light of the inferential statistics procedures that are paramount to the assignment. Such endeavor should help the writer/researcher to meet the goal of the theoretical basis for the assignment.
Correlation and Linear Frameworks
The correlation and linear patterns usually found in statistical analyses indicate that the role of independent and dependent variables is essential in the analysis of data as well as the levels of measurement utilized. In bivariate statistics and regression, as Berg (2004), and Myers, Gamst, and Guarino (2006) asserted, a flexibility of roles of the variables: playing one role in one context, and another role in another context can help explain their effects based on data collection methods used. This is important for the type of research design, the writer/researcher posits.
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A linear regression shows how a distribution is presented depending on the values of a variable x, and how another variable y varies. The relationship between these variables is the key concern. There is an effort to define a best line to ascertain the paths of the measures of central tendency (mean, variance, standard deviation…) (Berg, 2004, p.24). A simple linear equation is defined as y = a+ bx, where y is defined as the dependent variable, and x as the independent variable. The intercept, a constant, is labeled as a, and b the coefficient, is also considered as a slope. A bivariate relationship captured in a scatterplot shows how the relationship, and the shape of the bivariate between the variables are presented.
Statistical Basis
The focus of this assignment is generated from a researcher’s willingness to examine factors influencing reading scores among school children. 7 variables considered yield substantive descriptive statistics showing whether correlation relationships exist among the variables. The descriptive statistics in tables 1, 2, 3 and 4 SPSS below provide a complete picture of the variables, frequencies generated, and guided the writer/researcher description of the results. The descriptive statistics show that females have a higher frequency percentage (55.6 %) than males (24.4%). Reading rank has a higher mean than visual acuity, a lower standard deviation, and variances compared to visual acuity. This seems to suggest that the predictor and outcome variables can be considered in a bivariate domain, and correlation designs, as one variable relate to another; but there are other missing factors and further analysis to substantiate a valid conclusion or result (Keppel, Saufley, & Tokunaga, 1992).
The mean, standard deviation, median, and variances for reading rank are: 16.43 (mean), standard deviation (6.188), median (18), and variance (38.287), compared to visual acuity of 10.41 (mean), standard deviation (6.254), median (10.00) median, and 39.114(variances). For every standard deviation increase in visual acuity, there will be a 6.188 standard deviation in reading rank values.
Furthermore, for visual acuity rank, with a standard deviation of SD =6.254, skewness of .121, as depicted in table 5, and reading rank with SD =6.188, and skewness of -.965, suggesting both a skewness < 3.3 (statistics/standard error of skewness), the distribution seems to be normal. In figure 1 showing the output of a scatterplot matrix for visual acuity, and reading rank, the writer/researcher sees enough linearity due to a possible relationship between the variables when also carrying the Pearson’s correlation. However, further considerations and analysis would be needed since the scatterplot may not be too linear.
Descriptive Statistics
Table 1. Descriptive Statistics. Selected Variables
Variables
Group
Mean
Standard Deviation
Median
Frequencies (%)
Variances
Gender
1.56
.502
2.00
.252
Female
30 (55.6)
Male
24 (44.4)
Total
54 (100)
ID
12.94
6.534
13.00
42.645
1
1(1.9)
2
1 (1.9)
3
1 (1.9)
4
4 (7.4)
5
1 (1.9)
6
1 (1.9)
7
4 (7.4)
8
4 (7.4)
9
4 (7.4)
10
1 (1.9)
11
1 (1.9)
12
14 (1.9)
13
4 (7.4)
14
4 (7.4)
15
4 (7.4)
16
1 (1.9)
17
1 (1.9)
18
1 (1.9)
19
4 (7.4)
20
4 (7.4)
21
1 (1.9)
22
1 (1.9)
23
1 (1.9)
24
4 (7.4)
Total
54 (100)
Reading in Rank
16.43
6.188
18.00
38.287
1
1 (1.9)
2
1 (1.9)
3
1 (1.9)
4
1 (1.9)
5
1 (1.9)
6
1 (1.9)
7
1 (1.9)
8
1 (1.9)
9
1 (1.9)
10
1 (1.9)
11
1 (1.9)
12
2 (3.7)
13
1 (1.9)
14
1 (1.9)
15
4 (7.4)
16
4 (7.4)
17
4 (7.4)
18
4 (7.4)
19
4 (7.4)
20
4 (7.4)
21
4 (7.4)
22
4 (7.4)
23
4 (7.4)
24
4 (7.4)
Total
54 (100)
Visual Acuity in Rank
10.41
6.254
10.00
39.114
1
4 (7.4)
2
4 (7.4)
3
1 (1.9)
4
1 (1.9)
5
4 (7.4)
6
4 (7.4)
7
4 (7.4)
8
1 (1.9)
9
1 (1.9)
10
5 (9.3)
11
4 (7.4)
12
1 (1.9)
13
1 (1.9)
14
1 (1.9)
15
5 (9.3)
16
1 (1.9)
17
1 (1.9)
18
1 (1.9)
19
5 (9.3)
20
5 (9.3)
Total
54 (100)
Table 2 Descriptive Statistics
N
Minimum
Maximum
Mean
Std. Deviation
Variance
Gender
54
1
2
1.56
.502
.252
Visual Acuity Rank
54
1
20
10.41
6.254
39.114
ReadingRank
54
1
24
16.43
6.188
38.287
ID
54
1
24
12.94
6.534
42.695
Valid N (listwise)
54
Table 3.Visual Acuity Rank Frequencies
Frequency
Percent
Valid Percent
Cumulative Percent
Valid
1
4
7.4
7.4
7.4
2
4
7.4
7.4
14.8
3
1
1.9
1.9
16.7
4
1
1.9
1.9
18.5
5
4
7.4
7.4
25.9
6
4
7.4
7.4
33.3
7
4
7.4
7.4
40.7
8
1
1.9
1.9
42.6
9
1
1.9
1.9
44.4
10
5
9.3
9.3
53.7
11
4
7.4
7.4
61.1
12
1
1.9
1.9
63.0
13
1
1.9
1.9
64.8
14
1
1.9
1.9
66.7
15
5
9.3
9.3
75.9
16
1
1.9
1.9
77.8
17
1
1.9
1.9
79.6
18
1
1.9
1.9
81.5
19
5
9.3
9.3
90.7
20
5
9.3
9.3
100.0
Total
54
100.0
100.0
Table 4. Reading Rank Frequencies
Frequency
Percent
Valid Percent
Cumulative Percent
Valid
1
1
1.9
1.9
1.9
2
1
1.9
1.9
3.7
3
1
1.9
1.9
5.6
4
1
1.9
1.9
7.4
5
1
1.9
1.9
9.3
6
1
1.9
1.9
11.1
7
1
1.9
1.9
13.0
8
1
1.9
1.9
14.8
9
1
1.9
1.9
16.7
11
1
1.9
1.9
18.5
12
2
3.7
3.7
22.2
13
1
1.9
1.9
24.1
14
1
1.9
1.9
25.9
15
4
7.4
7.4
33.3
16
4
7.4
7.4
40.7
17
4
7.4
7.4
48.1
18
4
7.4
7.4
55.6
19
4
7.4
7.4
63.0
20
4
7.4
7.4
70.4
21
4
7.4
7.4
77.8
22
4
7.4
7.4
85.2
23
4
7.4
7.4
92.6
24
4
7.4
7.4
100.0
Total
54
100.0
100.0
Table 5. Descriptive Statistics and Skewness
N
Minimum
Maximum
Mean
Std. Deviation
Variance
Skewness
Statistic
Statistic
Statistic
Statistic
Statistic
Statistic
Statistic
Std. Error
Gender
54
1
2
1.56
.502
.252
-.230
.325
Visual Acuity Rank
54
1
20
10.41
6.254
39.114
.121
.325
Reading Rank
54
1
24
16.43
6.188
38.287
-.965
.325
ID
54
1
24
12.94
6.534
42.695
.066
.325
Valid N (listwise)
54
Figure 1. Scatterplot for visual Acuity and Reading Rank Matrix
Bivariate Statistic and Regression
Multiple Regression and P-values: a) Relationship Between Social Studies, Math,
and Reading. To ascertain the relationship between these variables, the writer/researcher
posits the following research questions, and hypotheses:
RQ1: What is the relationship between social studies, math, and reading considered concurrently?
Ho: There is no statistically significant relationship between social studies, math, and reading considered concurrently.
Halt: There is a statistically significant relationship between social studies, math, and reading considered concurrently.
SPSS correlations in table 1 results reveal a correlation of .342 between social studies and math, and .647 between social studies and reading. The analysis revealed a significant and positive correlation between math and reading. r=. 342, and p= .011 for math; and r=.647, with p = .000, p < 0.01 for reading. The null hypothesis is thus rejected for the alternative hypothesis.
Table 1. Correlations
Social Studies
Math
Reading
Social Studies
Pearson Correlation
1
.342*
.647**
Sig. (2-tailed)
.011
.000
N
54
54
54
Math
Pearson Correlation
.342*
1
.423**
Sig. (2-tailed)
.011
.001
N
54
54
54
Reading
Pearson Correlation
.647**
.423**
1
Sig. (2-tailed)
.000
.001
N
54
54
54
*. Correlation is significant at the 0.05 level (2-tailed).
**. Correlation is significant at the 0.01 level (2-tailed).
b) Partial correlation coefficients for variables reading and social studies with gender held as a constant.
RQ2: What is the relationship between reading and social studies with gender held constant?
Ho: There is no statisticallysignificant relationship between reading and social studies with gender held constant.
Halt: There is a statisticallysignificant relationship between reading and social studies with gender held constant.
A partial coefficient using bivariate regression analysis was conducted for the variable reading and social studies, holding the variable gender as constant. The analysis shows a positive correlation between social studies and reading r = .643, p= < .01. With a df (51), and p value < .01(statistically significant), the null hypothesis is rejected for the alternative hypothesis. Table 2 summarizes the analysis results.
Table 2 Partial Correlation Reading and Social Studies
Control Variables
Reading
Social Studies
Gender
Reading
Correlation
1.000
.643
Significance (2-tailed)
.
.000
df
0
51
Social Studies
Correlation
.643
1.000
Significance (2-tailed)
.000
.
df
51
0
c) Partial correlation coefficients for variables math and reading with gender held as a constant.
RQ3: Is there a relationship between math and reading with gender being held constant?
Ho: There is no statisticallysignificant relationship between math and reading with gender being held constant.
Ho: There is no statisticallysignificant relationship between math and reading with gender being held constant.
A SPSS analysis was conducted for the variables math and reading with gender held as a constant. The analysis showed a positive correlation with r =. 354, df (51), the correlation between math and reading was significant.p= .009 <0.01.
Table 3. Partial Correlation with Math and Reading
Control Variables
Math
Reading
Gender
Math
Correlation
1.000
.354
Significance (2-tailed)
.
.009
df
0
51
Reading
Correlation
.354
1.000
Significance (2-tailed)
.009
.
df
51
0
Appropriate Test to Measure Correlation Between Reading in Rank, and Visual
Acuity: A bivariatetest would be appropriate to measure the correlation between reading in rank, and visual acuity. SPSS analysis revealed a negative correlation between reading rank and visual acuity. r = -0.067, p =0.628 < 0.01 is not statistically significant.
Table 4. Reading in Rank and Visual Acuity Correlations
ReadingRank
Visual Acuity Rank
ReadingRank
Pearson Correlation
1
-.067
Sig. (2-tailed)
.628
N
54
54
Visual Acuity Rank
Pearson Correlation
-.067
1
Sig. (2-tailed)
.628
N
54
54
Linear Regressions
1) Social Studies as a Predictor of Reading Scores: Consistent with the notion of
interactive effects of variables, espoused by Agresti (2011), linear regressions
summarizing relationships between variables ought to make sense of data, and seem to suggest more than one predictor is needed for generalizing regression analyses (Berk, 2004, p.21).
A bivariate regression analysis was conducted for social studies as a predictor for social studies. R2 =.419, F (1, 52) =37.449, reading = 55.347, Social studies = 4.257,
and p < 0.01 as statistically significant. Tables 6 to 8 summarize analysis results. Table 9 also shows the bivariate regression for social studies, and reading.
Table 5 Variables Entered/Removeda
Model
Variables Entered
Variables Removed
Method
1
Social Studiesb
.
Enter
a. Dependent Variable: Reading
b. All requested variables entered.
Table 6. Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change
F Change
df1
df2
Sig. F Change
1
.647a
.419
.407
11.113
.419
37.449
1
52
.000
a. Predictors: (Constant), Social Studies
Table 7ANOVAa
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
4624.798
1
4624.798
37.449
.000b
Residual
6421.739
52
123.495
Total
11046.537
53
a. Dependent Variable: Reading
b. Predictors: (Constant), Social Studies
Table 8 Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
95.0% Confidence Interval for B
Collinearity Statistics
B
Std. Error
Beta
Lower Bound
Upper Bound
Tolerance
VIF
1
(Constant)
55.347
7.597
7.285
.000
40.102
70.591
Social Studies
4.257
.696
.647
6.120
.000
2.861
5.652
1.000
1.000
a. Dependent Variable: Reading
Table 9 Bivariate Regression for Social Studies as Predictor of Reading
Regression Weights
Variables b B
Social studies 4.257 .647
R2 .419
F 37.449
2) Math as a Predictor of Reading Scores: A bivariate regression was conducted for math and reading scores. Tables 11, 12, and 13 summarized analysis results. The bivariate regressions regression model with social studies as a predictor shows:
R2= .179
R2= .163 (adjusted), F (1, 52) =11.326 is significantly significant p=.001, and reading=74.549+1.155(math).
Table 11 below shows, B=.423 for math; thus for every standard deviation increase in math scores there is a .423 standard deviation rise in reading scores.
Table 10. Bivariate Regression for Math as a Predictor of Reading Scores
Regression weighs
Variables b B
Math 1.155 .423___________________________
Table 11 Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
Change Statistics
R Square Change
F Change
df1
df2
Sig. F Change
1
.423a
.179
.163
13.208
.179
11.326
1
52
.001
a. Predictors: (Constant), Math
Table 12 ANOVAa
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
1975.717
1
1975.717
11.326
.001b
Residual
9070.820
52
174.439
Total
11046.537
53
a. Dependent Variable: Reading
b. Predictors: (Constant), Math
Table 13 Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
95.0% Confidence Interval for B
Collinearity Statistics
B
Std. Error
Beta
Lower Bound
Upper Bound
Tolerance
VIF
1
(Constant)
74.549
8.036
9.277
.000
58.424
90.674
Math
1.155
.343
.423
3.365
.001
.466
1.844
1.000
1.000
a. Dependent Variable: Reading
The SPSS outputs are included for further review by the professor if needed.
References
Agresti, A. (2011). An introduction to categorical data analysis (2nd Ed). Hoboken, NJ: John Wiley & Sons.
Berk, R.A. (2004). Regression analysis. A constructive critique. Thousand Oaks, CA: Sage.
Keppel, G., Saufley, W.H., Jr., & Tokunaga, H. (1992). Introduction to design and analysis: A student’s handbook (2nd ed.). New York: W.H. Freeman.
Meyers, L. S., Gamst, G., & Guarino, A.J. (2006). Applied multivariate research. Design and interpretation. Thousand Oaks, CA: Sage.
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