

ORIGINAL ARTICLE 

Year : 2016  Volume
: 3
 Issue : 4  Page : 180189 

Sample size calculation to data analysis of a correlation study in Microsoft Excel^{®} : A handson guide with example
Himel Mondal^{1}, Shaikat Mondal^{2}
^{1} Department of Physiology, Maharaja Krishna Chandra Gajapati Medical College, Ganjam, Odisha, India ^{2} Department of Physiology, Medical College and Hospital, Kolkata, West Bengal, India
Date of Submission  13Oct2016 
Date of Decision  05Dec2016 
Date of Acceptance  15Dec2016 
Date of Web Publication  29Dec2016 
Correspondence Address: Himel Mondal Department of Physiology, MKCG Medical College, Ganjam  760 004, Odisha India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/23488832.196896
Background and Aim: Investigators use correlation study to determine the extent to which two or more variables are related among a single group of people. One big question frequently encountered by investigators is How to calculate sample size? After data collection, it is essential to test the data statistically. Hence, we planned to provide handson guide for sample size calculation and statistical analysis of data in Microsoft Excel^{®} 2010 commonly needed for a correlation study. Methods: An example of a correlation study of body mass index and peak expiratory flow rate was taken. Its execution process from sample size calculation to statistical analysis of data has been explained step by step in Excel^{®} 2010. All formula and functions were tested in Excel^{®} 2007. Conclusion: Excel^{®} as a tool for statistical analysis of data would facilitate researchers to do all statistical analyses in a single file. Along with correlation study, analysis by ttest and ANOVA was also provided. With this discussion, new investigators would get a fair knowledge about sample size calculation and data analysis of correlation study in Excel^{®} and can use it for their future studies. Keywords: Correlation study, Excel 2010, sample size, statistical analysis, ttest
How to cite this article: Mondal H, Mondal S. Sample size calculation to data analysis of a correlation study in Microsoft Excel^{®} : A handson guide with example. Int J Clin Exp Physiol 2016;3:1809 
How to cite this URL: Mondal H, Mondal S. Sample size calculation to data analysis of a correlation study in Microsoft Excel^{®} : A handson guide with example. Int J Clin Exp Physiol [serial online] 2016 [cited 2019 Mar 26];3:1809. Available from: http://www.ijcep.org/text.asp?2016/3/4/180/196896 
Introduction   
Correlation study
Correlation study is used to determine the relationship between two or more continuous variables among a single group of people. A correlation has direction, and correlation coefficient (r) can be either positive or negative. Zero indicates no relationship between the two variables, and r = 1 or r = −1 indicate a perfect relationship. The strength can be anywhere between 0 and ±1. ^{[1]} Correlation gives the degree and direction of relationship between the two variables, whereas the regression analysis predicts the values of one variable on the basis of the other variable. When there is perfect correlation (r = +1 or −1), the two regression lines will coincide or become one straight line. ^{[2]}
Essential terminology
Null hypothesis
The null hypothesis states that there is no association between the predictor and outcome variables in the population.
Alternative hypothesis
The proposition that there is an association is called the alternative hypothesis. The alternative hypothesis cannot be tested directly; it is accepted by default if the test of statistical significance rejects the null hypothesis.
Type I error
A type I error (false positive) occurs if an investigator rejects a null hypothesis that is actually true in the population.
Type II error
Type II error (false negative) occurs if the investigator fails to reject a null hypothesis that is actually not true in the population [Table 1]. ^{[3]}
Effect size
Effect size is a quantitative reflection of the magnitude of some phenomenon that is used for the purpose of addressing a question of interest. The investigator should first try to find data from prior studies in related areas to estimate reasonable effect size. When data are not available, a small pilot study may be done. ^{[4]}
α: The significance level of a test; the probability of rejecting the null hypothesis when it is true (or the probability of making a type I error).
β: The probability of failing to reject the null hypothesis when it is false (or the probability of making type II error).
Confidence level
The probability than an estimate of population parameter is within certain specified limits of the true value; commonly denoted by "1  α."
Power of test
The probability of correctly rejecting the null hypothesis when it is false; commonly denoted by "1  β." ^{[5]}
Materials and methods   
Commonly, researchers save raw data in Excel^{®} and do statistical analysis in other software. It may be beneficial for them if they can do statistical analysis in the same Excel^{®} sheet. Hence, we have discussed about the procedure of statistical analysis of data in Excel^{®} . Screenshots of Windows were captured by Greenshot 1.2.7 Build 2 software for Windows. Though the screenshots were taken while executing analysis in Excel^{®} 2010, we have tested all formula and functions in Excel^{®} 2007, and all functions and formulae were compatible with Excel^{®} 2007.
Designing a correlation study
Our research question was  whether body mass index (BMI) is correlated with peak expiratory flow rate (PEFR)? Similar previous studies with correlation of BMI and PEFR were available and had been studied beforehand. Among those studies, one study was almost similar regarding the age, race, and other characteristics of participants of our aimed study. That study found a correlation coefficient (r) = 0.30. Hence, we had anticipated in our study that BMI would be correlated with PEFR as that study.
Our null hypothesis
There is no correlation between BMI and PEFR in the study population.
Alternative hypothesis
There is a correlation between BMI and PEFR in the study population.
α: (two sided) we needed only 5% chance of rejecting null hypothesis when it is true. Hence, "α" value was 0.05. Confidence level was 1 − 0.05 = 0.95 or 95%.
β: We needed 5% probability of failing to reject null hypothesis when it is false. Hence, "β" value was 0.05 and power of test was 1 − 0.05 = 0.95 or 95%.
Effect size
From the previous study, we found correlation coefficient (r) = 0.30 between BMI and PEFR. Hence, our effect size was \0.30\ = 0.30.
The formula for sample size calculation for correlation study is as follows: ^{[6]}
N = ([Z_{α} + Z_{β}]/C) ^{2} + 3
where, N = number of subjects required
Z_{α} = the standard deviation for α
Z_{β} = the standard deviation for β
C = 0.5*ln ([1 + r]/[1 − r])
r = expected correlation coefficient
Sample size calculator
A new Excel^{®} sheet was opened. Three cells were selected for texts  "α=," "β=," and "expected r =." Immediate right adjacent cells were colored for easy identification. We used those cells for inserting values for α, β, and r. On the right side of those cells, a cell text was written as "Z_{α}=," and the adjacent right cells' formula was written as "=NORMSINV((1−(cell address of α/2)))." In our example, it was "=NORMSINV((1− (B4/2)))" [Figure 1]. As "α" was "two sided," we had divided it by 2. In similar fashion, "Z_{β}=" cells were selected and the formula was written as "=NORMSINV((1 − cell address of β))." In the example, it was "=NORMSINV((1 − B6))." Another cell text was written as "C=" and the adjacent right cells' formula was written as "=0.5*LN ((1 + cell address of expected r)/(1 − cell address of expected r))." In our example data sheet, it was "=0.5*LN ((1 + B8)/(1 − B8))" [Figure 2]. For the cell right side to "Sample size=," formula was written as "= (((cell address of Z_{α} + cell address of Z_{β})/cell address of C)^2) +3." In the said example, it was "= (((E4 + E6)/E8)^2) +3" [Figure 3]. For making the number as an integer, the formula is "=ROUND (cell address of sample size, 0)." In the example of [Figure 3], it was "=ROUND (E11, 0)." With this calculator, after inputting values of α, β, and r, we found that the number of participants required for our study was 139.  Figure 1: Screenshot showing calculation of Zƒ¿ in cell E4 with the value of cell B4 in Excel® 2010
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 Figure 3: Calculation of sample size for correlation study in cell E11 with the calculated value of cells E4, E6, and E8 based on the equation: N = ([Zƒ¿ + ZƒÀ]/C) 2 +3
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This calculator can be saved and used for sample size calculation of any correlation study with the effect size being calculated by correlation coefficient (r). Only α, β, and r values are required in specified cells, and the sample size will be calculated according to the formula we used.
Data entry
We had taken the following data (1) height in cm to the nearest 0.1 cm, (2) weight in kg to the nearest 0.1 kg, and (3) PEFR in L/min. In an Excel^{®} sheet, the 1 ^{st} column (column B) label was written as "Weight (kg)," in first row and in the first row, the 2 ^{nd} column label was written as "Height (cm)," the 3 ^{rd} as BMI, and the 4 ^{th} as "PEFR (L/min)," [Figure 4] and the data were entered accordingly. For ease of description, only eight sets of data were entered.  Figure 4: Calculation of body mass index in cell D2 with the value of cells B2 and C2 on the basis of Quetelet's equation: body mass index = Weight in kg/height in m^{2}
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Calculating body mass index, mean, standard deviation and standard error
For calculation of BMI according to Quetelet's equation, ^{[7]} we had to convert the "Height in cm" to "Height in meter." A cell below the "BMI" cell was selected and the formula was written as "=cell address of weight in kg/(cell address of height (cm)/100)^2." In the example of [Figure 4], it was "=B2/(C2/100)^2."
Arithmetic mean is the average. Hence, for the calculation of mean of the data of the column B, the cell equation was written as "=AVERAGE (starting cell address of column B: last cell address of column B)." For cell B10, it was "=AVERAGE (B2:B9)" [Figure 5]. For calculation of standard deviation (SD), the cell equation used with STDEV function was "=STDEV (starting cell address of column: Last cell address of column)." Here, it was "=STDEV (B2:B9)" in cell B11. For calculation of standard error (SE), we have to use the formula SE = SD√N, ^{[1]} where N is the sample size. If we use this formula in Excel^{®} for the data of column B in the example of [Figure 5], it will be "=STDEV (B2:B9)/SQRT ((COUNT (B2:B9)))." We can do it more simply as we already calculated the SD and we know the N. We can obtain the SE by writing a cell formula as "=B11/SQRT(8)" also.  Figure 5: Calculation of mean of values of cell B2 through B9 in the cell B10 with the "AVERAGE" function
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Calculation of Pearson's correlation coefficient (r)
For calculation of Pearson's correlation coefficient (r), we had written a cell text as "r=," and the formula for adjacent right cell was written with PEARSON function as: "=PEARSON (first cell address of first column: Last cell address of first column, first cell address of second column: Last cell address of second column)." Here, it was "=PEARSON (D2:D9, E2:E9)." In the example of [Figure 6], the value for "r" was calculated in cell H2 and the value was 0.675689.  Figure 6: Calculation of Pearson's correlation coefficient (r) of body mass index (column D) and peak expiratory flow rate (column E) data in cell H2 with the "PEARSON" function
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Alternatively, we could use the "Data Analysis Tools" to get the correlation coefficient. For correlation analysis, we have to follow the steps: Data tab (on ribbon)&$8594; Data Analysis Tools (on the extreme right side under data menu)&$8594;Correlation&$8594;Ok&$8594; Input range ($D$2:$E$9 for the example) and output range (any cell where the result to be shown, e.g., $A$13) &$8594; Ok.
Under "Data" tab (on ribbon), if "Data Analysis Tools" is not shown, we have to follow the steps: File &$8594; Option &$8594; Add Ins &$8594; Select "Analysis ToolPak" &$8594; Click on "Go" &$8594; Click "Ok" in new window [Figure 7]. Now, we have enabled data analysis tool for Excel^{®} . ^{[8]}  Figure 7: Serial screenshots showing from left to right: After clicking "File," "Options" to be clicked; in "Add‑Ins," "Analysis ToolPak" is to be selected and "Go" button to be clicked; "Analysis ToolPak" is ticked and "Ok" is to be clicked
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Calculation of "P value"
We needed to test whether the result was statistically significant or not. For calculation of "P value" from the "r" value, we had used the formula "=TDIST ((r*SQRT (n2))/(SQRT (1 − r^2)), n − 2,2)" for twotailed "P value." Calculation of "P0" from "r" is not allowed in Excel^{®} , so we had incorporated conversion of "r0" to "t" in the Excel^{®} formula according to the conversion formula: T = r(n − 2)/√(1 − r^{2} ) ^{[1]}
where r = correlation coefficient, n = number of participants, SQRT = square root function as written in Excel^{®} , last 2 in formula was for 2 tail P.
In our example, n = 8 and r = 0.675689, we had not put the value of "r," we had written the cell number of "r0" instead. In the example of [Figure 8], it was H2. We had written a cell text as "P value=," and on the right adjacent cell, we had written the formula as "=TDIST((H2*SQRT (6))/(SQRT (1H2^2)), 6, 2)." Calculated twotailed "P value" for the example data was 0.065879 in cell H4.  Figure 8: Calculation of two‑tailed "P value" in cell H4 with the help of calculated value of "r" of cell H2 with the Excel® function "TDIST"
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Solution for problems with negative "r" value: Where the "r0" value is negative, we have to convert it to its absolute value of number (ignoring the negative sign) before using it in the calculation of P from r. To do this in any cell with ABS function, we have to write the formula for the cell as "=ABS (cell address of r)." This will show the absolute value in the selected cell.
Solution for scientific expression of number: When the value of "P" is far less than zero (frequently encountered), the number in Excel^{®} is expressed in scientific format (e.g., 4.567E23). To show it in decimal format, we have to right click the cell where the number is present and have to select "Format Cells…." Under "Number" tab, we have to select "Number" (below "General") and increase the "Decimal places" until the number is shown after zeros in "Sample" and have to click "Ok." If the cell width is automatically increased to accommodate the number, it is ok. Otherwise, increase the width manually to view the number.
Regression analysis
In experimental science, after having understood the correlation between two variables, there are situations when it is necessary to do regression analysis. ^{[1]} For regression analysis, we had selected "Data&$8594; Data Analysis Tools&$8594;Regression&$8594;Ok&$8594; Input range and output range &$8594; Ok."
In the example, we put "BMI" in "X range" and "PEFR" in "Y range." For that, we had selected the "BMI" data column in "X range" as: "$D$2:$D$9" and "PEFR" data column in "Y range" as: "$E$2:$E$9" [Figure 9]. We could change the "Confidence Level," but it was 95% by default, so we kept it. We selected "Output range" in cell I1. The regression analysis result is shown in [Figure 10]. The equation y = α + βx is defined as the linear regression equation, where α is the intercept and β is the regression coefficient.  Figure 9: Screenshot during regression analysis of data of column D (Input Range X) and column E (Input Range Y) with 95% confidence interval. Output range is on cell I1
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 Figure 10: Summary output of regression analysis. Important parameters are colored. Equation: Y =145.75 + 3.972*X can be written from the data of J17 and J18
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In the example, we found: R^{2} = 0.4566; SE: 16.956; F: 5.041; significance of F: 0.0659; and the equation was "Y = 145.75 + 3.972*X."
The Student's ttest
The ttest is commonly used to determine whether the mean value of a continuous outcome variable in one group differs significantly from that in another group. ^{[9]} The ttest was not suitable for the study example we were discussing, but learning ttest procedure is a worth learning for any statistical analysis. Procedure is same as any "Data Analysis" we discussed above. We just need to select "ttest" from scrolling down after selecting "Data Analysis Tools" and input the range of data column in "Variable 1 Range" and "Variable 2 Range." In the example shown in [Figure 11], we wanted to carry out ttest for BMI of male (column A) and female (column B). We input "$A$2:$A$9" in "Variable 1 Range" and "$B$2:$B$9" in "Variable 2 Range." We have selected "$D$3" as output range. Values for tStat and P have been highlighted with color in the result of ttest in [Figure 11].  Figure 11: Screenshot of result of t‑test of column A and column B by using Data Analysis Tool (output range is D3). Calculation of only two‑tailed "P value" by Excel® function "TTEST" is shown in cell E1
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If we want to get only "P value" without other data analysis parameters, we can use the Excel^{®} function "TTEST" and write the formula as "=TTEST (A2:A9, B2:B9, 2, 2)" as shown in the example provided in [Figure 11]. Here, the "P value" has been calculated in cell E1. See the formula; after the array 1 and array 2, the first "2" is for twotailed test and the second "2" is for ttest assuming two samples of equal variance. We can change it according to our choice. If it is one tail and assuming two samples with unequal variance, the formula would be "=TTEST (A2:A9, B2:B9, 1, 3)."
ANOVA
In the example shown in [Figure 12], we have systolic blood pressure data of three groups of patients of underweight, normal weight, and overweight. We have to use the Data Analysis Tools for ANOVA test. Here, we have selected the "ANOVA: Single Factor" and selected input range as "$A:$C" and output range as $E$1. As our first row contained labels, we had ticked the "Labels in the first row." Result is shown on the sheet as shown in [Figure 12]. This example is for oneway ANOVA. For twoway ANOVA, we have to select "ANOVA: TwoFactor with Replication." For twoway ANOVA, we have to tell Excel^{®} about the "Rows per sample." Rest of the procedure is the same.  Figure 12: Analysis result of one‑way ANOVA for the data of columns A, B, and C (output range is E1)
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Regression scatter plot
Graphical representation of regression analysis data gives a better understanding of the trend. In the example of [Figure 13], we have shown a scatter plot of regression analysis of column A (BMI) and column B (Fat%) data. For inserting the scatter plot on Excel^{®} sheet, we have to click "Insert" tab on ribbon and in "Charts" section, "Scatter" to be clicked and "Scatter with only Markers" is to be selected. A blank graph plot will appear on the excel sheet. "Right click" on any area of the blank plot will give us option to "Select Data…." Clicking on "Select Data" will lead us to "Chart Data Range" selection. Column A and Column B have to be selected (Press and hold "Ctrl" and click on A and B column), and the "Chart Data Range" will appear as "=Sheet1!$A:$A, Sheet1!$B:$B." Alternatively, we can input the range as "=Sheet1!$A$1:$B$9" for the data in the example. Clicking on "Ok" will show the graph. Now, we have to right click on any series point (blue diamond) and select "Add Trend line…" and in "Trend line option," "Linear" is selected; "Display equation on chart" and "Display Rsquared value on chart" are selected and the window is closed. Regression scatter plot with trend line will be shown with equation and R2 value. In this example of data [Figure 13], BMI is on X axis and Fat% is on Y axis. The linear relation equation is "Y = 1.5948*X − 10.408."  Figure 13: Regression scatter plot for column A and B data, body mass index is on X axis and Fat% is on Y axis
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Discussion   
Adequate sample size has paramount importance in assessing the accurate results, while less or more than required sample size has many disadvantages. ^{[10]} Sample size must be calculated during the planning stage of the study. Moreover, it should be done with reference to previous studies or a pilot study. ^{[11]} There are websites which help in the calculation of sample size ^{[12],[13]} free of cost. However, the commonly used software for statistical analysis such as IBM SPSS Statistics (New Orchard Road Armonk, New York, USA) ^{[14]} and GraphPad Prism (GraphPad Software, Inc., CA, USA) ^{[15]} are paid software. It may be difficult for researchers in small settings to get that software. On the other hand, Microsoft Office^{®} is widely used software in educational institutions and offices. Excel^{®} is a part of Microsoft Office^{®} software package. Knowledge of Function, Formula, and Data Analysis Tools in Excel^{®} is necessary for the analysis of data. We have discussed a simple example of a correlation study for better understanding of the procedure of a correlation study.
Conclusion   
This article would help new researchers to calculate the sample size for correlation study and statistical analysis without using any additional statistics software. They get handson experience on making sample size calculator, calculation and analysis of data in Excel^{®} . Additional description of ttest and ANOVA would help them in data analysis of other studies. They can utilize this article as a reference tutorial for future studies.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12], [Figure 13]
[Table 1]
