Null and Alternative Hypothesis

• Hypothesis testing is a crucial decision-making process for evaluating population claims.
• It's a core part of statistical inference, used to validate a hypothesis with sample data.
• The process includes defining the population, stating hypotheses, setting significance levels, sampling, data collection, and statistical calculation to reach a conclusion (Bluman, 2009).
• The process starts with forming two competing hypotheses: the null and alternative.

Null Hypothesis vs. Alternative Hypothesis

• These concepts are foundational in statistical hypothesis testing to guide data analysis.
• The null hypothesis asserts no effect or relationship between variables.
• The alternative hypothesis (H₁) posits the existence of an effect or relationship.
• Testing aims to evaluate data evidence against the null hypothesis determine whether to reject it based on a significance level (alpha), denoted as (α) (Turner et al., 2020; Curran-Everett, 2009; Newman, 2008).
• In clinical trials, a null hypothesis might state that a new treatment has no difference from a standard one.
• Researchers focus on the null hypothesis to determine rejection using statistical tests (Huang et al., 2014).
• Rejection usually happens when the p-value is below a significance level of 0.05 (Graaf & Sack, 2018).
• The alternative hypothesis represents what researchers aim to support, crucial for understanding rejection implications.
• It can be one-sided (directional) or two-sided (non-directional), shaping result interpretations (Brereton, 2020).
• Acceptance of the alternative hypothesis suggests a meaningful effect or difference (Ratain & Karrison, 2007; Huang et al., 2014).

Two-Tailed Test Hypotheses

• Two-tailed tests involve a non-directional alternative hypothesis.
• It indicates the parameter of interest can be greater or less than a value.
• Null Hypothesis (H₀): μ = μ₀ (population mean equals a specified value)
• Alternative Hypothesis (H₁): μ ≠ μ₀ (population mean does not equal the specified value)
• Common in psychology and medicine to detect any difference from a known value (Montalbán et al., 2016; Turner et al., 2020).
• Preferred without prior expectations, offering a conservative approach (Prescott, 2019; Curran-Everett, 2009).

One-Tailed Test Hypotheses

• Uses directional alternative hypotheses.
• It suggests the parameter is either greater or less than a value.
• Null Hypothesis (H₀): μ ≤ μ₀ (population mean is less than or equal to a specified value)
• Alternative Hypothesis (H₁): μ > μ₀ (population mean is greater than the specified value)

Left-tailed test

• H₀: parameter = specific value
• H₁: parameter < specific value

Right-tailed test

• H₀: parameter = specific value
• Hₐ: parameter > specific value
• This approach applies when a direction of effect is expected based on prior research (Ruxton & Neuhäuser, 2010; Lombardi & Hurlbert, 2009).
• There is controversy regarding one-tailed tests, they can cause misinterpretations if the direction is wrong (Hurlbert & Lombardi, 2012; Serlin, 2000).
• Critics say one-tailed tests can inflate Type I error if misused (Ruxton & Neuhäuser, 2010; Lombardi & Hurlbert, 2009).
• Definitions use "parameter" but extend to distributions and randomness.

Clarifications: Signs in Hypothesis

• Two-tailed test: H₀ assumes population mean (μ) equals value (k), H₁ tests for any difference (μ ≠ k), rejection region in both distribution tails.
• Right-tailed test: H₀ assumes µ = k, H₁ suggests μ is greater than k (μ > k), rejection region is in the right tail.
• Left-tailed tests are similar, but H₁ suggests μ is less than k (μ < k), rejection region is in the left tail.

Common Phrases in Hypothesis Testing

• = means Is equal to
• = means Is the same as or Is exactly the same as
• > means Is increased or Is greater than/higher than
• ≥ means Is at least or Is not less than or Is greater than or equal to
• ≠ means Is not equal to, Is not the same, or Is different from
• < means Is decreased or Is less than/lower than
• ≤ means Is at most or Is not more than or Is less than or equal to

More Clarifications

• = is used in H₀ to indicate the parameter is exactly equal to a value.
• ≠ is used in H₁ for two-tailed tests, indicating the parameter is not equal to the value.
• > is used in H₁ for right-tailed tests, indicating the parameter is greater than the value.
• < is used in H₁ for left-tailed tests, indicating the parameter is less than the value.
• ≥ is used to express that the population parameter is at least a certain value in right-tailed tests.
• ≤ is used to express that the population parameter is at most a certain value in left-tailed tests.

Type I vs. Type II Error

• Type I: Rejecting a true null hypothesis, leads to a false positive.
• Type II: Failing to reject a false null hypothesis, leads to a false negative.
• Alpha (α) is probability of a Type I error.
• Alpha is conventionally set to 0.05, or 5% chance of falsely rejecting the null hypothesis (Vermeesch, 2011; Christley, 2010; Zerem, 2014).
• Type I errors can cause, unnecessary treatments (Jiménez-Gamero & Analla, 2023; Dash & Ali, 2023).
• Beta (β) is probability of Type II error.
• There is trade-off; reducing one type of error increases the other (Christley, 2010; Kelter, 2020; Mudge et al., 2012).
• Lowering significance (α) to minimize Type I may increase Type II (Zerem, 2014; Mudge et al., 2012).
• Power is 1 - β, probability of correctly rejecting a false null hypothesis (Christley, 2010; Dash & Ali, 2023).
• Balancing these errors is important, especially in areas such as clinical trials (Owusu-Ansah et al., 2016; Jiménez-Gamero & Analla, 2023).

Significance Level and P-Value in Hypothesis Testing

• Significance level (alpha), often 0.05, is the threshold to reject the hypothesis.
• If the p-value is less than 0.05, then the hypothesis is rejected
• P-value implies the compatibility of data with it, not its truth (Tsushima, 2022; Castelnuovo & Iacoviello, 2022; In & Lee, 2024).
• Second-generation p-value (SGPV) aims for nuanced significance by adding context (Hopkins, 2022; Lakens & Delacre, 2020).
• P-values are more informative when used with statistical measures (Castelnuovo & Iacoviello, 2022).
• Use a comprehensive interpretation of evidence instead of just "significant" or "not significant" (Leo & Sardanelli, 2020).
• P-values over 0.05 can be valuable (Kwak, 2023).
• Setting the significance level to 0.05 means accepting a 5% chance of incorrectly rejecting the null hypothesis.
• Lowering it (to 0.01) makes the test more stringent and reduces the likelihood of Type I error, but increases Type II error (Jamil, 2024).
• If α = 0.05; Then there is a 5% risk of concluding a difference exists when there is no actual difference. Meaning, 5% chance of rejecting a true null hypothesis/
• If α = 0.01; There is a 1% risk of concluding a difference exists when there is none.

Three Arbitrary Significance Levels

• Hypothesis testing is part of inferential statistics, for making inferences using sample data
• Significance level, a, is a predetermined factor for evaluating a hypothesis.
• The choice of significance level can affect test outcomes:
  • α = 0.10: Indicates a 10% risk of Type I error, this is used in exploratory research.
• α = 0.05: Implies a 5% chance of rejecting a true hypothesis, standard in fields that minimize false positives. This is often considered a standard in fields such as psychology and medicine, where it is crucial to minimize false positives while still maintaining adequate power to detect true effects.
• α = 0.01: Indicates a stringent 1% Type I error risk, this is employed in high-stakes research such as new medical guidelines.

Implications of Choosing a Significance Level

• Type I Error Rate: Corresponds to the significant level.
• Lower α reduces the likelihood of incorrectly rejecting the hypothesis.
• Type II Error Rate: Conversely, a lower significance level can increase the risk of Type II errors
• Statistical Power: Power is the probability of correctly rejecting a false null hypothesis.
• Lowering α decreases power unless the sample size is increased.

Understanding the P-Value

• The P-value tests for decisions on whether to reject or fail to reject the null hypothesis.
• Represents the probability of obtaining a test statistic at least as extreme.
• The smaller the p-value, the stronger the data against the null hypothesis.
  • p-value ≤ a: Reject the null hypothesis.
  • p-value > a: Fail to reject the null hypothesis.
• The P-value method for testing hypotheses differs from the traditional method (Critical Value Method using z-score).

Understanding the Critical-Value Method

• The Critical-Value Method compares the test statistic, calculated from sample data, to a critical value.
• This threshold represents the rejection criteria.
• The observed value of the statistic (sample observation) is compared to critical values (population observation).
• These critical values are expressed as standard z values.
• For instance, if we use a level of significance of 0.05, the size of the rejection region is 0.05.
• If the test is two-tailed; the rejection region is divided into two equal parts

Parametric vs Non-Parametric Tests

• Parametric tests assume data follows a distribution (normal).
• They are based on parameters.
• Non-parametric tests don't assume distribution, flexible for data not meeting assumptions.
• Assumes normality: Parametric tests assume that the data is approximately normally distributed.
• Use of parameters: These tests typically rely on parameters such as the population mean and standard deviation.
• Data requirements are that the data used in parametric tests must be continuous
• Parametric tests are more powerful.

Common Parametric Tests

• One-sample t-test is uses to compare the sample mean with a known value.
• Independent samples t-test compares 2 independent groups.
• Paired samples t-test compares 2 related groups
• One-way ANOVA compares 3+ independent groups.
• Pearson's Correlation measures the linear relationship between two continuous variables.

When to Use Parametric Tests:

• Data should be nearly normally distributed, continuos, with a large sample size to rely on data.

No-Parametric Tests

• Known as distribution free tests, do not assume a specific distribution for the data.
• Tests are more flexible and can be used when the data does not meet the assumptions required by parametric tests.

Key Features of Non-Parametric Tests:

• With no distributional assumptions, non-parametric tests do not require the data to follow a normal distribution.
• The tests are based on ranks or categories.
• Are less powerful due to less information used within the test.

Common Non-Parametric Tests:

• Mann-Whitney U the non-parametric equivalent of the independent samples t-test; compares the distributions of two independent groups.
• Wilcoxon Signed-Rank Test; compares the differences between paired data points.
• Kruskal-Wallis H Test compares the distributions of three or more independent groups.
• Spearman's Rank Correlation measures the relationship between 2 variables based on ranks.
• Chi-square Test assesses the association between 2 variables.

When to Use Non-Parametric Tests:

• Data isn't normally distributed, the data is ordinal, and the sample size is small.

Key Differences Between Parametric and Non-Parametric Tests:

• Assumptions of Parametric Tests are to assume normal distribution whereas Non-Parametric Tests have no assumption about the distribution.
• Data Type required for Parametric Tests is continuous data whereas Non-Parametric Tests can be Ordinal or nominal data.
• Test Statistics' Parametric Tests require Mean, standard deviation, t-value, F-value, whereas Non-Parametric Tests require Ranks, frequencies, chi-square, U- statistic.
• Power for Parametric Tests is, Generally, more powerful when whereas Non-Parametric Tests are Less powerful.
• Use Case; Parametric Tests are Used when data is approximately normal, and for larger sample sizes in contrast to Non-Parametric Tests.
• An Example of Parametric Tests is a One-sample t-test whereas Non-Parametric Tests is: Mann-Whitney U test.

When to Choose Parametric vs Non-Parametric Tests

• Normal Distribution: If your data is normally distributed parametric tests are usually the preferred choice as they are more powerful and efficient.
• If the data is not normally distributed, then non-parametric tests should be used.
• Data Type: If your data is continuous tests can be used if the conditions are met while if you data is ranked non0parametric test are used.
• Parametric tests typically require larger sample sizes

Steps for Hypothesis Testing Using The P-Value Method in SPSS

• State the Hypotheses and identify the claim.
• Set the Significance level
• Select the Appropriate Test and Calculate the Test Statistic
• Compute the p-value
• Make a decision
• Interpret the Result or make a conclusion