Correlation and Simple Linear Regression
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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.
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 Pvalues: 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. (2tailed) 
.011 
.000 

N 
54 
54 
54 

Math 
Pearson Correlation 
.342^{*} 
1 
.423^{**} 
Sig. (2tailed) 
.011 
.001 

N 
54 
54 
54 

Reading 
Pearson Correlation 
.647^{**} 
.423^{**} 
1 
Sig. (2tailed) 
.000 
.001 

N 
54 
54 
54 

*. Correlation is significant at the 0.05 level (2tailed). 

**. Correlation is significant at the 0.01 level (2tailed). 
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 (2tailed) 
. 
.000 

df 
0 
51 

Social Studies 
Correlation 
.643 
1.000 

Significance (2tailed) 
.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 (2tailed) 
. 
.009 

df 
0 
51 

Reading 
Correlation 
.354 
1.000 

Significance (2tailed) 
.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. (2tailed) 
.628 

N 
54 
54 

Visual Acuity Rank 
Pearson Correlation 
.067 
1 
Sig. (2tailed) 
.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. R^{2 }=.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/Removed^{a} 

Model 
Variables Entered 
Variables Removed 
Method 

1 
Social Studies^{b} 
. 
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 
.647^{a} 
.419 
.407 
11.113 
.419 
37.449 
1 
52 
.000 

a. Predictors: (Constant), Social Studies 

Table 7ANOVA^{a} 

Model 
Sum of Squares 
df 
Mean Square 
F 
Sig. 

1 
Regression 
4624.798 
1 
4624.798 
37.449 
.000^{b} 

Residual 
6421.739 
52 
123.495 

Total 
11046.537 
53 

a. Dependent Variable: Reading 

b. Predictors: (Constant), Social Studies 

Table 8 Coefficients^{a}


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
R^{2}^{ }^{.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:
R^{2}= .179
R^{2}= .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 
.423^{a} 
.179 
.163 
13.208 
.179 
11.326 
1 
52 
.001 

a. Predictors: (Constant), Math 

Table 12 ANOVA^{a}


Model 
Sum of Squares 
df 
Mean Square 
F 
Sig. 

1 
Regression 
1975.717 
1 
1975.717 
11.326 
.001^{b} 

Residual 
9070.820 
52 
174.439 

Total 
11046.537 
53 

a. Dependent Variable: Reading 

b. Predictors: (Constant), Math 

Table 13 Coefficients^{a} 

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 (2^{nd} 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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