Need a Topic/Title and Research Proposal outline in Power Point format.

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Week 2:Quantitative Data

Analysis

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Quantitative Research Data Analysis

Learning objectives:

1. Quantitative Data Analysis:

1.1 Statistics and Descriptive statistical analysis:

Statistics is concerned with the systematic collection of numerical data and its interpretation.

Descriptive statistics are used to describe the basic features of the data that have been collected in a

study. They provide simple summaries about the sample and the measures (e.g. mean, mode,

median, range, standard deviation etc). Together with simple graphics analysis, they form the basis

of virtually every quantitative analysis of data. It should be noted that with descriptive statistics no

conclusions can be extended beyond the immediate group from which the data was gathered.

Mean: The average value of the entire set of numbers. The Mean or average is probably the most

commonly used method of describing central tendency. To compute the mean all the values are

added up and divided by the number of values. For example, the mean or average quiz score is

determined by summing all the scores and dividing by the number of students taking the exam.

Example:

15, 20, 21, 20, 36, 15, 25, 15

The sum of these 8 values is 167, so the mean is 167/8 = 20.875.

Mode: The number that appears most often in a set of numbers. To determine the mode, you might

again order the scores as shown above, and then count each one. The most frequently occurring

value is the mode. In our example, the value 15 is the mode as it occurs most frequently (three

times).

Median: The middle value between the largest and smallest in a set of numbers. One way to

 Understand statistical terms related to quantitative research (confidence intervals and p-values)

 Learn the basic statistical tests

 Define sampling, data distribution and randomization

 Explain probability and non-probability sampling and describe the different types of each

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calculate the median is to list all scores in numerical order, and then locate the score in the center of

the sample. For example, if there are 500 scores in the list, score #250 would be the median. If we

order the 8 scores shown above, we would get:

Example:

15,15,15,20,20,21,25,36

There are 8 scores and score #4 and #5 represent the halfway point. Since both of these scores are

20, the median is 20. If the two middle scores had different values, they should be added and

divided by two to calculate the median.

Dispersion: Dispersion refers to the spread of the values around the central tendency. There are two

common measures of dispersion, the range and the standard deviation.

Range: The difference between the largest and smallest in a set of numbers i.e. the highest value

minus the lowest value. In our example distribution, the high value is 36 and the low is 15, so the

range is 36 - 15 = 21.

Standard deviation: A quantity expressing by how much the members of a group differ from the

mean value for the group.

1.2 Visual aid:

A set of data on its own is very hard to interpret. There is a lot of information contained in the data,

but it is hard to see. Eye-balling your data using graphs and exploratory data analysis is necessary

for understanding important features of the data, detecting outliers, and data which has been

recorded incorrectly.

Outliers

Outliers are extreme observations which are inconsistent with the rest of the data. The presence of

outliers can significantly distort some of the more formal statistical techniques, and hence there is a

high need for preliminary detection and correction or accommodation of such observations, before

further analysis takes place. Usually, a straight line fits the data well. However, the outlier “pulls”

the line in the direction of the outlier, as demonstrated in the lower graph in Figure 2. When the line

is dragged towards the outlier, the rest of the points then fall farther from the line that they would

otherwise fall on or close to. In this case the “fit” is reduced; thus, the correlation is weaker.

Outliers typically occur from an error including a mismarked answer paper, a mistake in entering a

score in a database, a subject who misunderstood the directions etc. The researcher should always

seek to understand the cause of an outlying score. If the cause is not legitimate, the researcher

should eliminate the outlying score from the analysis to avoid distorts in the analysis.

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Figure 1. A demonstration of how outliers can be identified using graphs

Figure 2. The two graphs above demonstrate data where no outliers are observed (top graph) and data where an outlier

is observed (bottom graph).

Data distribution (Langley & Perrie, 2014):

Data can be "distributed" (spread out) in different ways:

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Figure 3. Distribution of Data

The Normal Curve (Bell Curve):

The graph of the normal distribution depends on two factors i.e. the mean (M) and the standard

deviation (SD). The location of the center of the graph is determined by the mean of the

distribution, and the height and width of the graph is determined by the standard deviation. When

the standard deviation is large, the curve is short and wide; when the standard deviation is small, the

curve is tall and narrow. Normal distribution graphs look like a symmetric, bell-shaped curve, as

shown above. When measuring things like people's height, weight, salary, opinions or votes, the

graph of the results is very often a normal curve.

2. Statistical Analysis (Burns & Grove, 2005):

2.1 One-tailed versus two-tailed test:

One-tailed test: A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling

distribution, is called a one-tailed test. For example, suppose the null hypothesis states that the mean is less than or

equal to 10. The alternative hypothesis would be that the mean is greater than 10.

Two-tailed test: When using a two-tailed test, regardless of the direction of the relationship you

hypothesize, you are testing for the possibility of the relationship in both directions. For example,

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we may wish to compare the mean of a sample to a given value x using a t-test. Our null hypothesis

is that the mean is equal to x.

Figure 4. One- tailed and two-tailed test

2.2 Alpha level (p value)

In statistical analysis the researcher examines whether there is any significance in the results.

The acceptance or rejection of a hypothesis is based upon a level of significance – the alpha (a) level

This is typically set at the 5% (0.05) a level, followed in popularity by the 1% (0.01) a level

These are usually designated as p, i.e. p =0.05 or p = 0.01

So, what do we mean by levels of significance that the 'p' value can give us?

The p value is concerned with confidence levels. This states the threshold at which you are prepared to accept the

possibility of a Type I Error – otherwise known as a false positive – rejecting a null hypothesis that is actually true.

The question that significance levels answer is 'How confident can the researcher be that the results have not arisen

by chance?'

Note: The confidence levels are expressed as a percentage.

So if we had a result of:

p = 1.00, then there would be a 100% possibility that the results occurred by chance.

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p = 0.50, then there would be a 50% possibility that the results occurred by chance.

p = 0.05, then we are 95% certain that the results did not arise by chance

p = 0.01, then we are 99% certain that the results did not arise by chance.

Clearly, we want our results to be as accurate as possible, so we set our significance levels as low as

possible - usually at 5% (p = 0.05), or better still, at 1% (p = 0.01)

Anything above these figures, are considered as not accurate enough. In other words, the results are not significant.

Now, you may be thinking that if an effect could not have arisen by chance 90 times out of 100 (p = 0.1), then that

is pretty significant.

However, what we are determining with our levels of significance, is 'statistical significance', hence we are much

more strict with that, so we would usually not accept values greater than p = 0.05.

So when looking at the statistics in a research paper, it is important to check the 'p' values to find

out whether the results are statistically significant or not.

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Figure 5. Regions of rejection at 95% and 99% confidence interval

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Table 1

Statistical Symbols

Accessed: http://www.statisticshowto.com/statistics-symbols/

2.3 Statistical tests (Field, 2013)

There are a number of tests that can be used to analyse quantitative data, depending on what the researcher is

looking for, what data were collected and how the data were collected.

Below are a few of the most common tests used to analyse quantitative data:

t-Test

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A t-Test is used to compare whether two groups have different average values (for example, whether men and

women have different average heights).

A difference is more likely to be meaningful and “real” if:

(1) the difference between the averages is large

(2) the sample size is large

(3) responses are consistently close to the average values and not widely spread out (the standard deviation is

low).

Example where a t-Test can be used:

A researcher hypothesizes that individuals who are allowed to sleep for only four hours will score significantly

lower than individuals who are allowed to sleep for eight hours on a cognitive skills test. Sixteen participants are

invited into a sleep lab and are randomly assigned to two groups. One group sleeps for eight hours and the other

group sleeps for four hours. The next morning all participants complete the SCAT (Sam's Cognitive Ability Test).

The researcher wants to find out whether the average SCAT scores differ between the two groups.

 Independent Samples t-Test: The Independent Samples t- Test compares the means of two independent groups

in order to determine whether there is statistical evidence that the associated population means are significantly

different.

 Dependent t-test: The dependent t-test (also called the paired t-test or paired-samples t-test) compares the

means of two related groups to determine whether there is a statistically significant difference between these

means.

Correlation analysis

Correlation analysis is a statistical test used to study the strength of a relationship between two, numerically

measured, continuous variables (e.g. height and weight). A sample correlation coefficient is estimated (denoted r)

and can range in value from −1 to +1 and quantifies the direction and strength of the linear association between

the two variables.

As correlation is a measure of association, you can also think of the results in terms of effect size:

.00-.19: very weak.

.20-.39: weak.

.40-.59: moderate.

.60-.79: strong.

.80-1.0: very strong.

 Pearson product moment correlation: Pearson's correlation is used to test the linear relationship between at

least two continuous variables.

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 Spearman correlation: Spearman’s correlation evaluates the monotonic relationship between two continuous

or ordinal variables. In a monotonic relationship, the variables tend to change together, but not necessarily at a

constant rate. The Spearman correlation coefficient is based on the ranked values for each variable rather than the

raw data.

ANOVA (Analysis of Variance)

ANOVA is one of a number of tests (ANCOVA - analysis of covariance - and MANOVA - multivariate analysis

of variance) that are used to describe/compare a number of groups.

Examples of when you might want to test differences between groups:

 A group of individuals with elevated anxiety are trying three different interventions: Cognitive behavioral

therapy (CBT), Attention bias modification (ABM) and the Mindfulness-based therapy (MBT). You want to see

whether one therapy is more effective than the others.

 A manufacturer has two different processes to make light bulbs. They want to know if one process is better than

the other.

 Students from different colleges take the same exam. You want to see if one college outperforms the other.

One way and two way ANOVA

One-Way ANOVA has one independent variable (1 factor) with > 2 conditions

– conditions = levels = treatments

e.g., for a brand of cola factor, the levels are:

Coke, Pepsi, RC Cola

Two-Way ANOVA has 2 independent variables (factors)

– each can have multiple conditions

e.g. Two Independent Variables (IV’s)

– IV1: Brand; and IV2: Calories

– Three levels of Brand:

• Coke, Pepsi, RC Cola

– Two levels of Calories:

• Regular, Diet

*When a factor uses independent samples in all conditions, it is called a between subjects factor i.e. between-

subjects ANOVA

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*When a factor uses related samples in all conditions, it is called a within-subjects factor i.e. within-subjects

ANOVA (referred to as repeated measures).

ANCOVA - analysis of covariance

Analysis of covariance (ANCOVA) blends ANOVA and regression that allows to compare one variable in 2 or

more groups considering (or correcting for) variability of other variables, called covariates.

Retrieved from http://www.statsmakemecry.com/smmctheblog/stats-soup-anova-ancova-manova-mancova

MANOVA - multivariate analysis of variance

A MANOVA is an ANOVA with two or more continuous response variables. Like ANOVA, MANOVA has both

a one-way flavor and a two-way flavor. The number of factor variables involved distinguish a one-way

MANOVA from a two-way MANOVA.

Retrieved from http://www.statsmakemecry.com/smmctheblog/stats-soup-anova-ancova-manova-mancova

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MANCOVA

Both a MANOVA and MANCOVA feature two or more response variables, but the key difference between the

two is the nature of the IVs. While a MANOVA can include only factors, an analysis evolves from MANOVA to

MANCOVA when one or more covariates are added to the mix.

Retrieved from http://www.statsmakemecry.com/smmctheblog/stats-soup-anova-ancova-manova-mancova

Regression Analysis

In statistical modeling, regression analysis is a statistical process used to estimate the linear relationship between

two or more variables. It includes many techniques for modeling and analyzing several variables, when the focus

is on the relationship between a dependent variable and one or more independent variables (or 'predictors'). More

specifically, regression analysis shows how the typical value of the dependent variable changes when any one of

the independent variables is varied, while the other independent variables are held fixed.

Example scenario: Suppose you are a sales manager and want to predict next month’s numbers. A number of

factors from the weather to a competitor’s promotion to the rumor of a new and improved model can impact the

number of sales. e.g. The more the rain, the more the sales.”

Regression analysis is a way of mathematically sorting out which of those variables does indeed have an impact

on sales. It answers the questions: Which factors matter most? Which can we ignore? How do those factors

interact with each other? And, perhaps most importantly, how certain are we about all of these factors?

 Simple Regression: The simplest regression models involve a single response variable Y and a single predictor

variable X.

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 Multiple Regression: The Multiple Regression procedure fits a model relating a response variable Y to multiple

predictor variables X1, X2, ... . The user may include all predictor variables in the fit or ask the program to use a

stepwise regression to select a subset containing only significant predictors.

3. Parametric and Nonparametric Tests (Frost, 2015)

A parametric statistical test makes assumptions about the parameters (defining properties) of the

population distribution(s) from which one's data are drawn, whereas a non-parametric test makes no

such assumptions. Nonparametric tests are also called distribution-free tests because they do not

assume that your data follow a specific distribution.

It is argued that nonparametric tests should be used when the data do not meet the assumptions of the parametric

test, particularly the assumption about normally distributed data. However, there are additional considerations

when deciding whether a parametric or nonparametric test should be used.

3.1 Reasons to Use Parametric Tests

Reason 1: Parametric tests can perform well with skewed and nonnormal distributions

Parametric tests can perform well with continuous data that are not normally distributed if the

sample size guidelines demonstrated in the table below are satisfied.

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*Note: These guidelines are based on simulation studies conducted by statisticians at Minitab.

Reason 2: Parametric tests can perform well when the spread of each group is different

While nonparametric tests don not assume that your data are normally distributed, they do have other assumptions

that can be hard to satisfy. For example, when using nonparametric tests that compare groups, a common

assumption is that the data for all groups have the same spread (dispersion). If the groups have a different spread,

then the results from nonparametric tests might be invalid.

Reason 3: Statistical power

Parametric tests usually have more statistical power compared to nonparametric tests. Hence, they

are more likely to detect a significant effect when one truly exists.

3.2 Reasons to Use Nonparametric Tests

Reason 1: Your area of study is better represented by the median

The fact that a parametric test can be performed with nonnormal data does not imply that the mean is the best

measure of the central tendency for your data.

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For example, the center of a skewed distribution (e.g. income), can be better measured by the median where 50%

are above the median and 50% are below. However, if you add a few billionaires to a sample, the mathematical

mean increases greatly, although the income for the typical person does not change.

When the distribution is skewed enough, the mean is strongly influenced by changes far out in the distribution’s

tail, whereas the median continues to more closely represent the center of the distribution.

Reason 2: You have a very small sample size

If the data are not normally distributes and do not meet the sample size guidelines for the parametric tests, then a

nonparametric test should be used. In addition, when you have a very small sample, it might be difficult to

ascertain the distribution of your data as the distribution tests will lack sufficient power to provide meaningful

results.

Reason 3: You have ordinal data, ranked data, or outliers that you cannot remove

Typical parametric tests can only assess continuous data and the results can be seriously affected by

outliers. Conversely, some nonparametric tests can handle ordinal data, ranked data, without being

significantly affected by outliers.

4. Experimental Design (McLeod, 2007)

Experimental design refers to how participants are allocated to the different conditions (or IV levels) in an

experiment.

Three types of experimental designs are commonly used:

1. Independent Measures:

This type of design is also known as between groups. Different participants are assigned to a different condition

of the independent variable. This means that each condition of the experiment includes a different group of

participants. In this experimental design random allocation should be used, to ensure that each participant has

an equal chance of being assigned to one group or the other.

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Example:

2. Repeated Measures:

This type of design is also known as within subjects. The same participants are exposed to all the experimental

conditions (i.e. each condition of the experiment includes the same group of participants).

* Control: To control for order effects in repeated measures design the researcher counter balances the order of

the conditions for the participants i.e. alternating the order in which participants perform in different conditions

of an experiment.

Counterbalancing

Suppose that in a repeated measures design all of the participants first learned words in 'loud noise' and then

learned it in 'no noise'. We would expect the participants to show better learning in 'no noise' simply because

of order effects, such as practice (i.e. the same words are repeated in both conditions). A researcher can control

for order effects using counterbalancing.

Example:

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3. Matched Pairs:

Different participants are used in each condition, but participants are ‘matched’ as far as possible on relevant

variables in terms of any important characteristic which might affect performance, e.g. gender, age, intelligence

etc.

One member of each matched pair must be randomly assigned to the experimental group and the other to the

control group.

5. Power of the study:

There is increasing criticism about the lack of statistical power of published research in sports and exercise science

and psychology. Statistical power is defined as the probability of rejecting the null hypothesis; that is, the

probability that the study will lead to significant results. If the null hypothesis is false but not rejected, a type 2

error is incurred. Cohen suggested that a power of 0.80 is satisfactory when an alpha is set at 0.05—that is, the risk

of type 1 error (i.e. rejection of the null hypothesis when it is true) is 0.05. This means that the risk of a type 2 error

is 0.20.

The magnitude of the relation or treatment effect (known as the effect size) is a factor that must receive a lot of

attention when considering the statistical power of a study. When calculated in advance, this can be used as an

indicator of the degree to which the researcher believes the null hypothesis to be false. Each statistical test has an

effect size index that ranges from zero upwards and is scale free. For instance, the effect size index for a correlation

test is r; where no conversion is required. For assessing the difference between two sample means, Cohen's d ,

Hedges g, or Glass's Δ can be used. These divide the difference between two means by a standard deviation.

Formulae are available for converting other statistical test results (e.g. t test, one way analysis of variance, and χ2

results—into effect size indexes (see Rosenthal, 1991).

Effect sizes are typically described as small, medium, and large. Effect sizes of correlations that

equal to 0.1, 0.3, and 0.5 and effect sizes of Cohen's that equal 0.2, 0.5, and 0.8 equate to small,

medium, and large effect sizes respectively. It is important to note that the power of a study is

linked to the sample size i.e. the smaller the expected effect size, the larger the sample size required

to have sufficient power to detect that effect size.

For example, a study that assesses the effects of habitual physical activity on body fat in children might have a

medium effect size (e.g. see Rowlands et al., 1999). In this study, there was a moderate correlation between

habitual physical activity and body fat, with a medium effect size. A large effect size may be anticipated in a study

that assesses the effects of a very low energy diet on body fat in overweight women (e.g. see Eston et al, 1995). In

Eston et al’s study, a significant reduction in total body intake resulted in a substantial decrease in total body mass

and the percentage of body fat.

The effect size should be estimated during the design stage of a study, as this will allow the researcher to determine

the size required to give adequate power for a given alpha (i.e. p value). Therefore, the study can be designed to

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ensure that there is sufficient power to detect the effect of interest, that is minimising the possibility of a type 2

error.

Table 2.

Small, medium and large effect sizes as defined by Cohen

When empirical data are available, they can be used to assess the effect size for a study. However,

for some research questions it is difficult to find enough information (e.g. there is limited empirical

information on the topic or insufficient detail provided in the results of the relevant studies) to

estimate the expected effect size. In order to compare effect sizes of studies that differ in sample

size, it is recommended that, in addition to reporting the test statistic and p value, the appropriate

effect size index is also reported.

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References

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Surveys, Epidemiological Research, Programme Evaluation, Clinical Trials, Sixth Edition, 125-132.

Blaikie, N. (2003). Analyzing quantitative data: From description to explanation. Sage

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Burns N., Grove S.K. (2005). The Practice of Nursing Research: Conduct, Critique, and Utilization (5th Ed.). St.

Louis, Elsevier Saunders.

Creswell, J. W. (2013). Research design: Qualitative, quantitative, and mixed methods approaches. Sage

Publications, Incorporated.

Eston, RG, Fu F. Fung L (1995). Validity of conventional anthropometric techniques for estimating

body composition in Chinese adults. Br J Sports Med, 29, 52–6.

Field, A. (2013).Discovering Statistics Using IBM SPSS Statistics. (4th Ed).

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http://blog.minitab.com/blog/adventures-in-statistics-2/choosing-between-a-nonparametric-test-and-a-parametric-

test

Langley C, Perrie Y (2014). Maths Skills for Pharmacy: Unlocking Pharmaceutical Calculations. Oxford

University Press.

Lyons, R. (2010). Best Practices in Graphical Data Presentation. Ohio, USA.

Saul McLeod (2007). Simply Psychology. Retrieved from https://www.simplypsychology.org/experimental-

designs.html

Rosenthal R. (1991.). Meta-analytic procedures for social research (revised edition). Newbury Park, CA: Sage

Rowlands A.V, Eston R.G, Ingledew D.K. (1999). The relationship between activity levels, body fat and aerobic

fitness in 8–10 year old children. J Appl Physiol, 86, 1428–35.