hypothesis h1 h2 h3

5. METHODOLOGY 5.1 Hypotheses The research questions presented informally in Chapters 2 and 3 (see pp. 19, 22, 29, 30, 39, & 42) are recast below as formal hypotheses underlying the present research. The order coincides with the earlier discussions. H1 : Correlation between MT and ID is independent of model (Fitts, Welford, or Shannon) for index of difficulty H2 : Correlation between MT and ID is the same using ID calculated from the effective target width (in 1-D or 2-D) rather than the specified target width H3 : MT is independent of approach angle H4 : Error rate is independent of approach angle H5 : Correlation between MT and ID is independent of several methods of calculating W in the index of difficulty H6 : Movement time is independent of device or task H7 : Error rate is independent of device or task H8 : Index of performance is independent of device or task H9 : Point-drag-select tasks can be modeled as two separate Fitts' law tasks Hypotheses H1 , H2 , H6 , H7 , and H8 will be tested in the first experiment, H3 , H4 , and H5 in the second experiment, and H9 in the third. If the hypotheses were ranked by importance, H1 and H5 would stand out. The main contributions in this research lie in extending the theoretical and empirical arguments for the Shannon formulation over the Fitts' formulation ( H1 ), and in introducing a useful technique for extending Fitts' law to two dimensional target acquisition tasks ( H5 ). H1 has already been tested in the re-analysis of data from experiments by Fitts and others. Although the analyses provide evidence to reject the hypothesis (with the Shannon formulation out-performing the Fitts formulation), Experiment 1 is an attempt to provide first-hand empirical evidence. The extension of Fitts' law to dragging tasks ( H6 - H9 ) is important in light of the current genre of direct manipulation interfaces employing a variety of State 1 (pointing) and State 2 (dragging) transactions. Testing for device differences across tasks is important to aid in the design of direct manipulation systems employing pointing and dragging operations. H9 , tested in Experiment 3, is the weakest hypotheses in a statistical sense since rejecting a null hypothesis is not the basis of the analysis. Demonstration is the main objective. When a State 2 action follows a State 1 action (as commonly occurs), it is claimed that the effect is that of two Fitts' law tasks in sequence. Two prediction equations should apply, reflecting the inherent information processing capacities in each task. 5.2 Experiment 1 In Fitts' original experiment, subjects moved a stylus back and forth between two targets and tapped on them as quickly and accurately as possible (see Figure 1). Experiment 1 mimicked Fitts' paradigm except subjects manipulated an input device to move a cursor between two targets displayed on a CRT display. 5.2.1 Subjects Twelve male students in a computer engineering technology programme at a local college volunteered as subjects and were paid an hourly rate. All subjects used computers on a daily basis. Experience with input devices (other than a keyboard) was minimal. 5.2.2 Apparatus An Apple Macintosh II microcomputer (running system software version 6.0.4) served as the host computer with input through a mouse (standard equipment), a trackball, and a pressure-sensing tablet-with-stylus. The output display was a 33 cm colour CRT monitor (used in black-and-white mode) with a resolution of 640 by 480 pixels. The trackball was a Turbo Mouse ADB Version 3.0 by Kensington Microware Ltd. Both the mouse and the trackball contained a momentary switch – a "button" – which was pressed and released to select targets. The tablet, a Model SD42X by Wacom Inc., sensed pressure and absolute x-y coordinates using a stylus as input. The pressure of the stylus was sensed with 6 bits of resolution. Positioning was absolute: If the stylus was raised more than 1/4 inch above the tablet and then repositioned, the cursor jumped to the new x-y position when the stylus was re-engaged. If the stylus remained within 1/4 inch of the tablet, the cursor continued to track as the stylus moved. The control-display gain for the tablet-with-stylus was matched to that for the mouse which in turn was set to .53 (equivalent to the "fast" setting in the Macintosh control panel). Although the tablet-with-stylus had the potential for pseudo-analogue applications (with 6 bits of resolution), the present experiment used a threshold value set between 0 and 63 to signal a "select" operation. A threshold of 50 was chosen based on the subjective preferences of experienced users. 5.2.3 Procedure Subjects performed multiple trials on two different tasks using three different devices. The operation of the devices and the requirements of the tasks were explained and demonstrated to each subject before beginning. One warm-up block of trials was given prior to data collection. The two tasks were "point-select" and "drag-select". For the point-select task, subjects moved the cursor back and forth between the targets and selected each target by pressing and releasing a button on the device (or applying and releasing pressure on the stylus). An arrow pointing up appeared just below the target to be selected. After each selection, the arrow moved to the opposite target, thereby guiding subjects through the block of trials (see Figure 14a). Further feedback was provided by a black rectangle which appeared across the top of the display while in State 2. This feedback was particularly important with the stylus to inform subjects of a state change. (a) (b) Figure 14. Sample conditions for Experiment 1 showing (a) the point-select task and (b) the drag-select task

Figure 15 . Sample condition for Experiment 2

Figure 16. Sample condition for Experiment 3

Stats Hypothesis testing

Hypothesis testing

Hypothesis testing is one of the most widely used approaches of statistical inference.

The idea of hypothesis testing (more formally: null hypothesis significance testing - NHST) is the following: if we have some data observed, and we have a statistical model, we can use this statistical model to specify a fixed hypothesis about how the data did arise. For the example with the plants and music, this hypothesis could be: music has no influence on plants, all differences we see are due to random variation between individuals.

The null hypothesis H0 and the alternative hypothesis H1

Such a scenario is called the null hypothesis H0. Although it is very typical to use the assumption of no effect as null-hypothesis, note that it is really your choice, and you could use anything as null hypothesis, also the assumption: “classical music doubles the growth of plants”. The fact that it’s the analyst’s choice what to fix as null hypothesis is part of the reason why there are are a large number of tests available. We will see a few of them in the following chapter about important hypothesis tests.

The hypothesis that H0 is wrong, or !H0, is usually called the alternative hypothesis, H1

Given a statistical model, a “normal” or “simple” null hypothesis specifies a single value for the parameter of interest as the “base expectation”. A composite null hypothesis specifies a range of values for the parameter.

If we have a null hypothesis, we calculate the probability that we would see the observed data or data more extreme under this scenario. This is called a hypothesis tests, and we call the probability the p-value. If the p-value falls under a certain level (the significance level $\alpha$) we say the null hypothesis was rejected, and there is significant support for the alternative hypothesis. The level of $\alpha$ is a convention, in ecology we chose typically 0.05, so if a p-value falls below 0.05, we can reject the null hypothesis.

Test Statistic

Type I and II error

Significance level, Power

Misinterpretations

A problem with hypothesis tests and p-values is that their results are notoriously misinterpreted. The p-value is NOT the probability that the null hypothesis is true, or the probability that the alternative hypothesis is false, although many authors have made the mistake of interpreting it like that \citep[][]{Cohen-earthisround-1994}. Rather, the idea of p-values is to control the rate of false positives (Type I error). When doing hypothesis tests on random data, with an $\alpha$ level of 0.05, one will get exactly 5\% false positives. Not more and not less.

Further readings

• The Essential Statistics lecture notes
• http://www.stats.gla.ac.uk/steps/glossary/hypothesis_testing.html

Examples in R

Recall statistical tests, or more formally, null-hypothesis significance testing (NHST) is one of several ways in which you can approach data. The idea is that you define a null-hypothesis, and then you look a the probability that the data would occur under the assumption that the null hypothesis is true.

Now, there can be many null hypothesis, so you need many tests. The most widely used tests are given here.

The t -test can be used to test whether one sample is different from a reference value (e.g. 0: one-sample t -test), whether two samples are different (two-sample t -test) or whether two paired samples are different (paired t -test).

The t -test assumes that the data are normally distributed. It can handle samples with same or different variances, but needs to be “told” so.

t-test for 1 sample (PARAMETRIC TEST)

The one-sample t-test compares the MEAN score of a sample to a known value, usually the population MEAN (the average for the outcome of some population of interest).

Our null hypothesis is that the mean of the sample is not less than 2.5 (real example: weight data of 200 lizards collected for a research, we want to compare it with the known average weights available in the scientific literature)

t-test for 1 sample (NON-PARAMETRIC TEST)

One-sample Wilcoxon signed rank test is a non-parametric alternative method of one-sample t-test, which is used to test whether the location (MEDIAN) of the measurement is equal to a specified value

Create fake data log-normally distributed and verify data distribution

Our null hypothesis is that the median of x is not different from 1

Two Independent Samples T-test (PARAMETRIC TEST)

Parametric method for examining the difference in MEANS between two independent populations. The t -test should be preceeded by a graphical depiction of the data in order to check for normality within groups and for evidence of heteroscedasticity (= differences in variance), like so:

Reshape the data:

Now plot them as points (not box-n-whiskers):

The points to the right scatter similar to those on the left, although a bit more asymmetrically. Although we know that they are from a log-normal distribution (right), they don’t look problematic.

If data are not normally distributed, we sometimes succeed making data normal by using transformations, such as square-root, log, or alike (see section on transformations).

While t -tests on transformed data now actually test for differences between these transformed data, that is typically fine. Think of the pH-value, which is only a log-transform of the proton concentration. Do we care whether two treatments are different in pH or in proton concentrations? If so, then we need to choose the right data set. Most likely, we don’t and only choose the log-transform because the data are actually lognormally distributed, not normally.

A non-parametric alternative is the Mann-Whitney-U-test, or, the ANOVA-equivalent, the Kruskal-Wallis test. Both are available in R and explained later, but instead we recommend the following:

Use rank-transformations, which replaces the values by their rank (i.e. the lowest value receives a 1, the second lowest a 2 and so forth). A t -test of rank-transformed data is not the same as the Mann-Whitney-U-test, but it is more sensitive and hence preferable (Ruxton 2006) or at least equivalent (Zimmerman 2012).

To use the rank, we need to employ the “formula”-invokation of t.test! In this case, results are the same, indicating that our hunch about acceptable skew and scatter was correct.

(Note that the original t -test is a test for differences between means, while the rank- t -test becomes a test for general differences in values between the two groups, not specifically of the mean.)

Cars example:

Test the difference in car consumption depending on the transmission type. Check wherever the 2 ‘independent populations’ are normally distributed

Graphic representation

We have two ~normally distributed populations. In order to test for differences in means, we applied a t-test for independent samples.

Any time we work with the t-test, we have to verify whether the variance is equal betwenn the 2 populations or not, then we fit the t-test accordingly. Our Ho or null hypothesis is that the consumption is the same irrespective to transmission. We assume non-equal variances

From the output: please note that CIs are the confidence intervales for differences in means

Same results if you run the following (meaning that the other commands were all by default)

The alternative could be one-sided (greater, lesser) as we discussed earlier for one-sample t-tests

If we assume equal variance, we run the following

Ways to check for equal / not equal variance

1) To examine the boxplot visually

2) To compute the actual variance

There is 2/3 times difference in variance.

3) Levene’s test

Mann-Whitney U test/Wilcoxon rank-sum test for two independent samples (NON-PARAMETRIC TEST)

We change the response variable to hp (Gross horsepower)

The ‘population’ of cars with manual transmission has a hp not normally distributed, so we have to use a test for independent samples - non-parametric

We want to test a difference in hp depending on the transmission Using a non-parametric test, we test for differences in MEDIANS between 2 independent populations

Our null hypothesis will be that the medians are equal (two-sided)

Wilcoxon signed rank test for two dependend samples (NON PARAMETRIC)

This is a non-parametric method appropriate for examining the median difference in 2 populations observations that are paired or dependent one of the other.

This is a dataset about some water measurements taken at different levels of a river: ‘up’ and ‘down’ are water quality measurements of the same river taken before and after a water treatment filter, respectively

The line you see in the plot corresponds to x=y, that is, the same water measuremets before and after the water treatment (it seems to be true in 2 rivers only, 5 and 15)

Our null hypothesis is that the median before and after the treatment are not different

the assumption of normality is certainly not met for the measurements after the treatment

Paired T-test for two dependend samples test. (PARAMETRIC)

This parametric method examinates the difference in means for two populations that are paired or dependent one of the other

This is a dataset about the density of a fish prey species (fish/km2) in 121 lakes before and after removing a non-native predator

changing the order of variables, we have a change in the sign of the t-test estimated mean of differences

low p ->reject Ho, means are equal

Testing for normality

The normal distribution is the most important and most widely used distribution in statistics. We can say that a distribution is normally distributed when: 1) is symmetric around their mean. 2) the mean, median, and mode of a normal distribution are equal. 3) the area under the normal curve is equal to 1.0. 4) distributions are denser in the center and less dense in the tails. 5) distributions are defined by two parameters, the mean and the standard deviation (sd). 6) 68% of the area of a normal distribution is within one standard deviation of the mean. 7) Approximately 95% of the area of a normal distribution is within two standard deviations of the mean.

Normal distribution

Load example data

Visualize example data

Visual Check for Normality: quantile-quantile plot

This one plots the ranked samples from our distribution against a similar number of ranked quantiles taken from a normal distribution. If our sample is normally distributed then the line will be straight. Exceptions from normality show up different sorts of non-linearity (e.g. S-shapes or banana shapes).

Normality test: the shapiro.test

As an example we will create a fake data log-normally distributed and verify the assumption of normality

Correlations tests

Correlation tests measure the relationship between variables. This relationship can goes from +1 to -1, where 0 means no relation. Some of the tests that we can use to estimate this relationship are the following:

-Pearson’s correlation is a parametric measure of the linear association between 2 numeric variables (PARAMETRIC TEST)

-Spearman’s rank correlation is a non-parametric measure of the monotonic association between 2 numeric variables (NON-PARAMETRIC TEST)

-Kendall’s rank correlation is another non-parametric measure of the associtaion, based on concordance or discordance of x-y pairs (NON-PARAMETRIC TEST)

Compute the three correlation coefficients

Test the null hypothesis, that means that the correlation is 0 (there is no correlation)

When we have non-parametric data and we do not know which correlation method to choose, as a rule of thumb, if the correlation looks non-linear, Kendall tau should be better than Spearman Rho.

Further handy functions for correlations

Plot all possible combinations with “pairs”

To make it simpler we select what we are interested

Building a correlation matrix

• Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student??????s t-test and the Mann-Whitney U test. Behavioral Ecology, 17, 688-690.
• Zimmerman, D. W. (2012). A note on consistency of non-parametric rank tests and related rank transformations. British Journal of Mathematical and Statistical Psychology, 65, 122-44.
• http://www.uni-kiel.de/psychologie/rexrepos/rerDescriptive.html

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• Knowledge Base
• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis (H 0 ): There’s no effect in the population .
• Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question
• They both make claims about the population
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
• Alternative hypothesis (H A ): Independent variable affects dependent variable .

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

The actual test begins by considering two  hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis : It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a  decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in  H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

In a  hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

• OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

Hypothesis Testing

• First Online: 01 October 2020

Cite this chapter

• Benito Damasceno 2  

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This chapter deals with the one-tailed and two-tailed testing of the null (Ho) hypothesis versus the experimental (H1) one. It describes types of errors (I and II) and ways to avoid them; limitations of α significance level in reporting research results as compared to confidence interval and effect size, which does not depend on sample size and is useful in meta-analysis studies; the value of pre-establishing a large enough sample size and sufficient statistical power for avoiding type I and II errors, particularly in clinical trials with new kinds of interventions; calculation of sample size; the problems of dropouts and small samples; and the contribution that can be given by studies with small samples or even single cases (e.g., in rare conditions as autism) using appropriate designs as the ABAB.

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Benito Damasceno

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Damasceno, B. (2020). Hypothesis Testing. In: Research on Cognition Disorders. Springer, Cham. https://doi.org/10.1007/978-3-030-57267-9_16

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Hypothesis testing

• PMID: 8900794
• DOI: 10.1097/00002800-199607000-00009

Hypothesis testing is the process of making a choice between two conflicting hypotheses. The null hypothesis, H0, is a statistical proposition stating that there is no significant difference between a hypothesized value of a population parameter and its value estimated from a sample drawn from that population. The alternative hypothesis, H1 or Ha, is a statistical proposition stating that there is a significant difference between a hypothesized value of a population parameter and its estimated value. When the null hypothesis is tested, a decision is either correct or incorrect. An incorrect decision can be made in two ways: We can reject the null hypothesis when it is true (Type I error) or we can fail to reject the null hypothesis when it is false (Type II error). The probability of making Type I and Type II errors is designated by alpha and beta, respectively. The smallest observed significance level for which the null hypothesis would be rejected is referred to as the p-value. The p-value only has meaning as a measure of confidence when the decision is to reject the null hypothesis. It has no meaning when the decision is that the null hypothesis is true.

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6. Operationalisation, Research Design, and Methodology

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1 This chapter focuses on the methodological criteria guiding the empirical testing of the three hypotheses introduced earlier (H1, H2, H3). For each of them, we first account for the operationalisation of related theoretical concepts, accounting for their chosen indicators. Second, we move to discussing aspects of research design, making sense of our selection of cases and units of analysis. Third, we move to addressing questions related to inference, also acknowledging potential threats to the internal validity of empirical conclusions, and seeking to control for them. Fourth, we consider measurement, construct-validity, and reliability issues. The above four steps are replicated in sequence for H1, H2, and H3.

6.1 Hypothesis 1

2 H1: If salience is manifested in relation to an FTA chapter when negotiations are ongoing (independent variable), political ratification obstacles (PROs) will be posed by veto players after signing (dependent variable).

Concept operationalisation

3 With respect to the independent variable, we regard i) salience as the explicit (a) manifestation of concerns that the way an FTA chapter is expected to address a given issue-area is undesired or insufficient. Salience can also take the form of (b) open criticism of the expected terms of an agreement, resulting from the anticipation of a utility loss on the part of would-be veto players. (Thomson and Stokman 2006: 41-42) As for the dependent variable, we conceive ii) PROs as the (a) manifested intention to veto an FTA on the part of EU institutional actors (Council, EP, individual Member States) or as (b) an invitation to reject an FTA by actors lacking direct veto powers (NGOs, unions, business groups). PROs are also found in the (c) open criticism to an FTA after signing , even if the intention to resort to a veto is not explicitly voiced. 1

4 The testing of H1 relies on a multiple-case parallel design, “in which several instrumental bounded cases are selected to develop a more in-depth understanding of the phenomena than a single case can provide” (Chmiliar 2012: 583) . We run empirical observations over six different trade-policy domains. These have been isolated and selected, on the one hand, as issue-areas representative of the “regulatory turn” characteristic of new-generation FTAs, for they go beyond an exclusive tariff approach to market integration (Meunier and Czesana 2019) . On the other hand, these are among the sectors the literature has identified as particularly contentious in post-Lisbon EU trade, and emerge as particularly politicised based on our preliminary exploration of recent FTAs (Duina 2019) . Thus, their contentiousness makes them relevant for the purpose of H1. Examined cases qualify as “parallel” for they have been selected a priori, based on their expected theoretical pertinence (Chmiliar 2012) . Examined policy domains are: i) Investment protection; ii) Trade and sustainable development (TSD); iii) Trade and labour; iv) Regulation and NTBs; v) Sanitary and phytosanitary (SPS) measures; vi) Geographical indications (GIs). 2

5 The units of analysis in each case are (would-be) veto players. At an official level, we considered the Council, the EP, and individual Member Governments. Private-sector veto players include advocacy NGOs, business organisations, social partners like trade unions, as well as the European Economic and Social Committee (EESC), all officially engaging with the Commissions in trade-policy auditing. Private veto players and social partners were purposely sampled via the EU Transparency Register, relying on research filters to ensure representative samples of stakeholders for each of the above six cases (European Commission 2020b) .

6 Our units of observation consist of statements, position papers, and reports issued by the above categories of actors. For the independent variable, one document (if available) from each institution or organisation in the sample was selected, and coded dichotomously as salience: 1/0 (i.e. “yes/no”) for each of the six cases examined. 3 The same process was repeated under the dependent variable, coding as PROs: 1/0 (i.e. “yes/no”). We performed our coding through the software Nvivo, which allows for the systematic organisation and numerical analysis of textual material, by means of its matrix-query function. Numerical results from the qualitative coding of textual entries are displayed in Table I below.

Table I: Numerical results of the empirical testing of H1 (expressed in percentage). Salience (1/0) constitutes the independent variable; PROs (1/0) the dependent variable. Cases in which salience was associated with PROs are highlighted in red. Control cases are highlighted in green.

7 In testing H1, we control for the possibility of a spurious relationship on three grounds. First, time order between cause and effect is ensured. Documents under the independent variable pre-date those coded to the dependent variable (Buttolph Johnson et al. 2016: 170) . Second, the criterion of covariation is met by two of the examined cases (Regulation and GIs) (169) . Covariation allows us to observe that not only is the presence of salience associated with PROs, but also its absence or low occurrence is associated with no or low PROs. Third, we seek to address threats to internal validity. In this respect, we acknowledge how, in principle, nonprobability purposeful sampling implies selection bias. Yet, when it comes to specialised interest groups engaging in EU trade dialogues on given issue-areas, the population is small enough to be examined directly with adequate accuracy, without relying on probabilistic generalisations (Trochim and Donnelly 2008: 48-49) . For instance, in testing the emergence of PROs related to trade and labour, we sampled all labour-related organisations officially accredited to consult with the Commission (largely unions). Thus, for the sake of sampling, we are not interested in whether labour issues are salient to EU stakeholders in general, but rather whether stakeholders working on each examined sector are observed to have posed PROs. Finally, we acknowledge that the inclusion of six cases for empirical testing qualifies as a small-N design. Yet, an in-depth approach to their analysis allows us to improve the testing of related inferences. As Trochim (1985) observes, the in-depth analysis of a case, and variables within it, can help control for threats to internal validity. Specifically, in-depth case studies run a fine-grained analysis of the causal mechanisms of interest, allowing for “pattern specificity” making possible alternative explanations less likely (580).

Construct validity and reliability

8 The consistency and systematicity of textual observations under H1 are improved by relying on “concept mapping,” which allows for the “translation of an idea or construct into something real and concrete” (Trochim and Donnelly 2008: 56). This seeks to ensure construct validity at three main levels. First, as discussed above, operative definitions of the theoretical concepts in H1 have been narrowly devised, looking at previous theories in political science that focus on issues of salience and propose specific empirical proxies for this notion (Thomson and Stokman 2006) . Second, a number of specific textual indicators have been attached to each operationalised concept, in order to account for the type of entries included in our coding. Third, the replicability of our measurements has been ensured by handing the textual material to a second coder, who re-coded the entire dataset. The resulting inter-coder reliability (ICR) was calculated by looking at the percentage of coding entries on which the author and the second coder agreed (Feng 2014) , and amounts to 85%.

6.2 Hypothesis 2

9 H2(a): If official-level veto players mobilise against the terms of an FTA following negotiations (independent variable), the Commission will resort to solution-chasing (dependent variable).

10 H2(b): If private/advocacy veto players mobilise against the terms of an FTA following negotiations (independent variable), the Commission will resort to solution-chasing (dependent variable).

11 In both H2(a) and H2(b), i) veto-player mobilisation encapsulates the notion of PROs, conceived as the (a) manifested intention to veto an FTA on the part of EU institutional actors (Council, EP, individual Member States); (b) an invitation to reject an agreement by private/advocacy veto players lacking direct ratification powers (NGOs, unions, business organisations); (c) open criticism to an FTA ahead of ratification , even if the intention to resort to a veto is not explicitly voiced. Further, mobilisation can also refer to (d) prolonged contestation of the terms of an FTA in the aftermath of ratification. The dependent variable is kept constant across both H2(a) and H2(b). The concept of ii) solution-chasing is operationalised both as the (a) adoption of additional safeguards or documents complementing an FTA; and as the (b) setting up of institutional fora in charge of carrying out monitoring and review ex post (e.g. DAGs) . Both types of instruments are usually implemented in the context of specific FTA chapters.

12 The testing of H2 relies on the qualitative comparative analysis (QCA) of post-negotiation debates on specific FTAs. QCA designs look at “configurations of causal factors understood as sufficient to produce an outcome” and display whether the occurrence/absence of one or more of these factors impacts on the realisation of the dependent variable (Gerring 2012: 343; see also Ragin 1987) . We focus on the six issue-areas previously examined in H1, this time making reference to a specific FTA in each of them. 4 In running our QCA, we define two causal variables expected to interact in determining whether the Commission triggers solution-chasing or not: the i) mobilisation of official-level veto players after signing; the ii) mobilisation of private/advocacy veto players after signing. Each variable is coded dichotomously: 1/0 (i.e. “yes/no”). In building our dataset, we relied on multiple data collection, looking at primary documents, news sources, and personal semi-structured interviews with both Commission officials and non-governmental stakeholders (Buttolph Johnson et al. 2016: 196) . Our design is displayed in Table II below.

Table II. Table II displays coding results related to the effect of the mobilisation of veto players (independent variables of H2) on the expected outcome of solution-chasing (dependent variable of H2). In the case of TSD within the EU-Mercosur FTA, we coded solution-chasing as a question mark since the agreement was only concluded in June 2019, and still awaits going through ratification. In the case of “Trade and Labour” (KOREU) the variables mobilisation of official-level veto players and solution-chasing were coded both as 0 and 1 since their change is observable across time. Specifically, when the former variable was absent (0) so was the latter. Likewise, when the former was present (1) so was the latter.

13 The employment of QCA is motivated by two main analytical and methodological objectives. First, in studying the phenomenon of solution-chasing, we go beyond seeking to establish a causal relationship between a single variable X and Y, rather accounting for the “multiple configurations of factors” concurring with the causal mechanism of interest (Gerring 2012: 333) . To that end, QCA can be conceived as a methodological attempt to reconcile the rationale of multivariate analysis with the qualitative observation of small samples of cases. Second, QCA is especially compatible with the comparative approach between our six selected cases. Specifically, assessing which independent variables occur (and do not occur) when solution-chasing is verified can help draw conclusions as to what actors and conditions can effectively push the Commission to revise the terms of a negotiated FTA – and which ones cannot.

14 QCA still shows its limitations when it comes to validating causal inferences. First, QCA lacks random selection, as data are chosen in light of their high representativeness of the causal phenomenon of interest. Second, our QCA relies on a restricted dataset. In general, that implies a small-N problem, by which the number of observations would be insufficient for standard multivariate analysis to statistically corroborate an inference. Yet, QCA can still effectively address small-N issues, by running the maximum number of comparisons between variables across selected cases (Ragin 1987) . This implies that, while not testing the impact of multiple variables over one outcome across a large-N, QCA looks at what combinations of these variables account for the outcome of interest. That is crucial to our research, allowing us to refine our understanding of the drives behind solution-chasing. Furthermore, QCA does a better job at controlling for possible alternative explanations than a simple (and potentially spurious) association between X and Y (Trochim 1985: 580) . In this light, although a small-N problem remains, the in-depth exploration of patterns of solution-chasing can still prove a helpful starting point for future more extensive and systematic research.

15 With respect to the independent variables, measurements of the mobilisation of official-level and private/advocacy veto players are guided by the documentary dataset relied upon for H1. In using the same dataset, the testing of H2 indirectly benefits from the inter-coder reliability (amounting to 85%) of measurements under H1. This also ensures the consistency of observations across the two hypotheses. Furthermore, the inclusion of additional instances of veto-player mobilisation following FTA ratification (and hence not covered by H1) is grounded on information gathered through semi-structured interviews with both Commission DG Trade officials and EU civil-society advocates. As for the dependent variable, the measurement of solution-chasing is aided by the fact that this concept of interest, and related indicators, consist of policy processes, and of the (non-) occurrence of a number of verifiable facts (e.g. implementation of safeguards, setting-up of advisory groups). This poses fewer measurement issues than in the presence of verbal statements potentially subject to multiple and inconsistent interpretations (cf. H1).

6.3 Hypothesis 3

16 H3(a): If a mixed FTA is provisionally enforced by unanimity (independent variable), lock-in dynamics will prevent national parliaments from formally rejecting the agreement (dependent variable).

17 H3(b): If the Commission involves private veto players in the provisional implementation of a mixed FTA (independent variable), lock-in dynamics will disincentivise these actors from lobbying for the rejection of the agreement (dependent variable).

18 H3(a): With respect to the independent variable, i) unanimous enforcement refers to the unanimous approval by the Council of the provisional enforcement of a mixed FTA. As for the dependent variable, ii) lock-in dynamics are conceived as involving national political majorities which, albeit critical of a provisionally enforced mixed FTA, nevertheless abstain from casting a veto as part of Member-State ratification.

19 H3(b): Under the independent variable, i) involvement in provisional implementation is conceived as the inclusion of private/advocacy veto players by the Commission into institutional implementation structures tasked with the monitoring and review of specific FTA chapters being provisionally implemented (e.g. DAGs). As for the dependent variable, ii) lock-in dynamics show multiple empirical configurations. They can take the form of (a) institutional incentives and (b) epistemic incentives binding private veto players to continue participating in these implementation structures. Institutional incentives can be found in how close and compatible the positions of partaking advocates are, which in turn points at how conductive the institution they belong to is towards advocacy coalition-building (Sabatier and Weible 2007) . Epistemic incentives are seen as indirectly proportional to the number of trade advocates monitoring FTA implementation on behalf of each member organisation in the DAG, and hence on their degree of dependency on information-sharing with fellow coalition members. The fewer the personnel working on trade in each partaking organisation, the higher its epistemic incentives to remain part of a given implementation structure ceteris paribus. Finally, lock-ins can consist of (c) resource incentives , both in terms of funding and personnel, making implementation structures set up by the Commission cost-efficient in the eyes of partaking veto players – the latter facing limited financial and human resources. 5

20 The design of H3(a) and H3(b) is grounded on an in-depth analysis of the single case study of CETA’s provisional enforcement. We deem the CETA as theoretically relevant for exploring policy dynamics affecting mixed FTAs. First, the CETA constitutes the only instance of a comprehensive mixed agreement provisionally enforced by the EU and yet to complete ratification by Member States. This FTA went to ratification ahead of ECJ Opinion 2/15, qualifying as shared competence investment-protection and portfolio-investment chapters, which ever since the EU-Singapore FTA have been negotiated separately from main texts (European Court of Justice 2017) . Nearly three years into CETA provisional enforcement are hardly sufficient to draw final conclusions as to the lock-in dynamics affecting the implementation of the FTA. Yet, a preliminary assessment can be made, based on existing empirics, as to the dynamics deemed conducive to longer-term path dependency (Pierson 2000) . Second, the CETA is particularly representative of the political contestation and ratification obstacles examined by this research (see H1). To a greater extent than other contemporary FTAs, the CETA has been at the centre of intense political debates challenging its ratification, often as a result of spill-overs from the TTIP (Duina 2019) .

21 H3(a): In accounting for lock-in dynamics in H3(a), we rely on a crucial-case least-likely design , running observations over one unit of analysis a priori unexpected to mirror the theoretical mechanism of interest. If a least-likely case turns out to comport with a theory, it can serve as a relevant empirical validation – to a greater extent than a priori neutral or likely cases (Gerring 2012: 234; Jordan 1999) . In light of this, we select the case of Italy as the only country having experienced a significant parliamentary-majority shift since the CETA was provisionally enforced by the Council (European Parliament 2019b) . Most crucially, the incumbent parliamentary majority has proven highly critical of the CETA, unlike the previous leadership which gave its consent to provisional enforcement. Thus, Italy is a priori least likely not to reject the CETA as part of national ratification.

22 H3(b): To examine lock-in dynamics in H3(b), we selected as our units of analysis private interest and advocacy groups sitting in the CETA DAG. H3(b) is therefore tested through a single-case small-N design. The narrow operationalisation of concepts under H3 has been carried out in view of accounting for multiple “theoretically defined classes of events” within the same case of interest (Levy 2008: 2) . Specifically, the distinction between (a) institutional incentives; (b) epistemic incentives; and (c) resource incentives allows us to test the occurrence of the dependent variable through narrowly defined proxies pointing at the outcome of interest.

23 H3(a): As observed by Eckstein (1975) , crucial-case least-likely designs are particularly suitable for theory-testing, providing that theoretical predictions are defined carefully. The internal validity of H3(a) benefits from the narrow definition of the theoretical mechanism linking unanimous enforcement (independent variable) to national-level lock-ins (dependent variable). Specifically, unanimity, while constituting the independent variable, also acts as a scope condition narrowing down the focus of H3(a) to unanimous Council decisions on trade. Further, the criterion of FTA “mixedness” also explicitly restricts the population of relevant agreements. Conditions for the selection of the unit of analysis (i.e. individual Member States) are also narrowly defined. In this regard, the case of Italy is selected based on two specific criteria. First, the country is yet to initiate its process of ratification of the CETA. Second, the incumbent parliamentary majority in Rome, at the time of writing, differs from the preceding political leadership which agreed on provisional enforcement in the Council. Thus, the narrow conditions defined above can improve the internal validity of inferences under H3(a), in the face of small-N problems in our design. As Gerring (2007) observes, “[t]he more a theory attains the status of a causal law, the easier it will be to confirm, or disconfirm, with a single case” (117).

24 H3(b): Analogously to H3(a), the testing of H3(b) employs a single-case design, exploring lock-in dynamics involving private/advocacy veto players sitting in the CETA DAG. In order to improve the internal validity of our inferences, we tackle the small-N problem faced by this design by running as many empirical observations as possible over the case of the CETA DAG. This is particularly important in single-case designs, where observations across multiple cases cannot be performed, and therefore need to be maximised within a single one. To that end, relying on three different indicators of the concept of lock-in (dependent variable) helps in refining the causal mechanism of interest from multiple perspectives. In that regard, simply looking at veto players’ nominal membership in the DAG would per se tell nothing about the extent to which these actors are dependent on (or locked into) CETA implementation structures. Conversely, examining how lock-ins can take the form of i) institutional incentives, ii) epistemic incentives , and ii) resource incentives offers a more accurate exploration of why veto players are bound to prolongedly contribute to a trade regime they still regard as suboptimal.

25 As regards construct validity, both the testing of H3(a) and H3(b) and their single-case approach are particularly suitable for ensuring measurement accuracy. Specifically, statistical limitations related to small-N problems can be compensated for by closely examining dynamics within a single case. In-depth analysis, in turn, requires careful and detailed operationalisation of theoretical concepts. To that end, indicators for H3 ensure construct validity to the extent that their link to their respective concepts has been theory-driven. For instance, in H3(b), we measure institutional incentives by looking at mandate proximity between DAG advocates. This is dictated by advocacy-coalition theory emphasis on how homogeneous positions among advocates incentivise them to act collectively, allowing them to obviate coordination problems (Sabatier and Weible 2007) . Finally, the fact that the concept of lock-in is operationalised through multiple and specific indicators strengthens reliability, allowing single empirical measurements to be less prone to misinterpretations and more easily replicable.

Notes de bas de page

1 Indicators employed for the coding under H1 are exemplified in Annex I.

2 We selected relevant data produced in the context of the KOREU (2010), TTIP (negotiations interrupted in 2016), CETA (2016), EU-Japan EPA (2018), EU-Singapore FTA (2019), EU-Mercosur FTA (2019), EU-Vietnam FTA (2019), and EU-Australia FTA (negotiations ongoing since 2018). Dates refer to the year of signing.

3 Only one document per institution or organisation (if available) was coded under the independent and dependent variable, in order not to inflate the number of textual entries.

4 See Table II. For TSD, empirics were gathered both in regard to the CETA and the EU-Mercosur FTA.

5 See Annex II for the complete dataset pertaining to institutional, epistemic, and resource incentives, relied upon in the examined case study of the CETA DAG (chapter VIII).

Le texte seul est utilisable sous licence Creative Commons - Attribution - Pas d'Utilisation Commerciale - Pas de Modification 4.0 International - CC BY-NC-ND 4.0 . Les autres éléments (illustrations, fichiers annexes importés) sont « Tous droits réservés », sauf mention contraire.

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Coloc: sensitivity to prior values

Chris wallace, sensitivity analysis.

Specifying prior values for coloc.abf() is important, as results can be dependent on these values. Defaults of \(p_1=p_2=10^{-4}\) seem justified in a wide range of scenarios, because these broadly correspond to a 99% belief that there is true association when we see \(p<5\times 10^{-8}\) in a GWAS. However, choice of \(p_{12}\) is more difficult. We hope the coloc explorer app will be helpful in exploring what various choices mean, at a per-SNP and per-hypothesis level. However, having conducted an enumeration-based coloc analysis, it is still helpful to check that any inference about colocalisation is robust to variations in prior values specified.

Continuing on from the last vignette , we have

A sensitivity analysis can be used, post-hoc, to determine the range of prior probabilities for which a conclusion is still supported. The sensitivity() function shows this for variable \(p_{12}\) in the bottom right plot, along with the prior probabilities of each hypothesis, which may help decide whether a particular range of \(p_{12}\) is valid. The green region shows the region - the set of values of \(p_{12}\) - for which \(H_4 > 0.5\) - the rule that was specified. In this case, the conclusion of colocalisation looks quite robust. On the left (optionally) the input data are also presented, with shading to indicate the posterior probabilities that a SNP is causal if \(H_4\) were true. This can be useful to indicate serious discrepancies also.

Let’s fake a smaller dataset where that won’t be the case, by increasing varbeta:

Now, colocalisation is very dependent on the value of \(p_{12}\) :

In this case, we find there is evidence for colocalisation according to a rule \(H_4>0.5\) only for \(p_{12} > 10^{-6}\) , which corresponds to an a priori belief that \(P(H_4) \simeq P(H_3)\) . This means but you would need to think it reasonable that \(H_4\) is equally likely as \(H_3\) to begin with to find these data convincing.

Note, the syntax can also consider more complicated rules:

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Last years' talks

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The Winter talks: 2018 code{white-space: pre;} pre:not([class]) { background-color: white; } if (window.hljs) { hljs.configure({languages: []}); hljs.initHighlightingOnLoad(); if (document.readyState && document.readyState === “complete”) { window.setTimeout(function() { hljs.initHighlighting(); }, 0); } } h1 { font-size: 34px; } h1.title { font-size: 38px; } h2 { font-size: 30px; } h3 { font-size: 24px; } h4 { font-size: 18px; } h5 { font-size: 16px; } h6 { font-size: 12px; } .

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Hypothesis testing (H1, H2, and H3).

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