Correlation and Simple Linear Regression

✅ Paper Type: Free Essay	✅ Subject: Data Analysis
✅ Wordcount: 10079 words	✅ Published: 18th May 2020

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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.

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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.

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/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

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		.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.