Hypothesis - Entry: Concepts & Abstractions

Interdiscipline Academy Theoretical Case Studies

A hypothesis is essentially an educated guess or a proposed explanation for a particular phenomenon or observation. It's a tentative idea that's put forward as a starting point for further investigation.

Think of it as a question you're asking about the world. You don't have the answer yet, but you have some preliminary thoughts based on what you already know or have observed. These thoughts are formulated into a testable statement – the hypothesis.

For example, let's say you notice that plants in your garden seem to grow taller when you water them more frequently. You might formulate a hypothesis: "If I increase the amount of water given to plants, then their growth rate will also increase."

This hypothesis suggests a relationship between two things: the amount of water and the plant's growth. It's not just a random guess; it's based on your observation.

The crucial aspect of a hypothesis is that it must be testable. You need to be able to design an experiment or study that can either support or disprove your initial idea. In our plant example, you would need to set up a controlled experiment where you give different amounts of water to different groups of plants and then carefully measure their growth over time.

If your experiment supports your hypothesis, it strengthens the evidence for your proposed explanation. However, if the results contradict your hypothesis, it doesn't necessarily mean it's wrong. It simply means that your initial idea needs to be revised or refined.

1. Z-test (for population mean with known standard deviation)

• Formula: z = (x̄ - μ) / (σ / √n)

  • where:

    • x̄ is the sample mean

    • μ is the population mean

    • σ is the population standard deviation¹

    • n is the sample size²

• Use Case: When you know the population standard deviation (σ) and have a large sample size (generally n ≥ 30).

2. T-test (for population mean with unknown standard deviation)

• Formula: t = (x̄ - μ) / (s / √n)

  • where:

    • x̄ is the sample mean

    • μ is the population mean

    • s is the sample standard deviation

    • n is the sample size

• Use Case: When you don't know the population standard deviation (σ) or have a small sample size (n < 30).

3. Chi-Square Test (for categorical data)

• Formula: χ² = Σ [(O - E)² / E]

  • where:

    • O is the observed frequency

    • E is the expected frequency

• Use Case:

  • To test the independence of two categorical variables.

  • To compare observed frequencies with expected frequencies in a single categorical variable.

In formal logic, a hypothesis is the antecedent of a proposition. This means it's the "if" part of an "if-then" statement. For example, in the proposition "If it rains, then the ground gets wet," "it rains" is the hypothesis.

The hypothesis is the assumed condition upon which the consequent (the "then" part) is dependent. It's a crucial element in deductive reasoning, where conclusions are drawn logically from premises.

1. ANOVA (Analysis of Variance)

F=MSwithin​MSbetween​​

where:

• F is the F-statistic

• MS_between is the mean square between groups

• MS_within is the mean square within groups

Mean squares are calculated by dividing the sum of squares by the corresponding degrees of freedom.

2. Regression Analysis

• Simple Linear Regression:

  Y=β0​+β1​X+ϵ

  where:

  • Y is the dependent variable

  • X is the independent variable

  • β₀ is the population intercept

  • β₁ is the population slope

  • ε is the error term

• Multiple Regression:

  Y=β0​+β1​X1​+β2​X2​+...+βp​Xp​+ϵ

  where:

  • Y is the dependent variable

  • X₁, X₂, ..., X_p are the independent variables

  • β₀ is the population intercept

  • β₁, β₂, ..., β_p are the population slopes

  • ε is the error term

3. Logistic Regression

P(Y=1)=1+e−(β0​+β1​X1​+β2​X2​+...+βp​Xp​)1​

where: * P(Y=1) is the probability of the event occurring (Y=1) * X₁, X₂, ..., X_p are the independent variables * β₀ is the intercept * β₁, β₂, ..., β_p are the coefficients

4. Survival Analysis

• Kaplan-Meier Estimator:

  S^(t)=ti​≤t∏​(1−ni​di​​)

  where:

  • S^(t) is the estimated survival probability at time t

  • t_i are the observed event times

  • d_i is the number of events at time t_i

  • n_i is the number of individuals at risk at time t_i

• Cox Proportional Hazards Model:

  h(t∣X)=h0​(t)e(β1​X1​+β2​X2​+...+βp​Xp​)

  where:

  • h(t|X) is the hazard function at time t given covariates X

  • h₀(t) is the baseline hazard function

  • X₁, X₂, ..., X_p are the covariates

  • β₁, β₂, ..., β_p are the coefficients

The null hypothesis, often denoted as H₀, is a fundamental concept in statistical hypothesis testing. It essentially proposes that there is no significant difference or relationship between the variables or groups being studied. In essence, it assumes that any observed effect is due to chance or random variation rather than a true underlying cause.

The null hypothesis serves as a starting point for statistical analysis. By formulating and testing the null hypothesis, researchers can determine whether their data provides sufficient evidence to reject it in favor of an alternative hypothesis, which suggests a meaningful effect or relationship.

For example, if a researcher is investigating the effectiveness of a new drug, the null hypothesis might state that there is no difference in the average health outcomes between patients receiving the drug and those receiving a placebo. The researcher then collects data and conducts statistical tests to determine whether the observed differences in health outcomes are statistically significant enough to reject the null hypothesis and conclude that the drug is indeed effective.

It's crucial to understand that failing to reject the null hypothesis does not necessarily mean it is true. It simply means that the available data does not provide sufficient evidence to conclude that it is false.

ANCOVA Equation (Simplified)

Y_ij = μ + τ_i + β(X_ij - X̄) + ε_ij

• Y_ij: The dependent variable for the j-th observation in the i-th group.

• μ: The overall mean of the dependent variable.

• τ_i: The effect of the i-th group.

• β: The regression coefficient for the covariate.

• X_ij: The value of the covariate for the j-th observation in the i-th group.

• X̄: The overall mean of the covariate.

• ε_ij: The error term for the j-th observation in the i-th group.

MANOVA, or Multivariate Analysis of Variance, is a statistical technique used to compare the means of multiple dependent variables simultaneously across different groups. It extends the concept of ANOVA, which examines the differences in a single dependent variable, to situations where you have several interrelated outcomes.

For example, if you're studying the effects of different teaching methods on student performance, you might use MANOVA to assess whether the methods impact not only students' test scores (one dependent variable) but also their classroom participation and attitudes towards the subject (other dependent variables).

Instead of conducting separate ANOVAs for each dependent variable, MANOVA considers the relationships between these variables, making it more powerful and sensitive to detecting group differences.

• Wilks' Lambda: This statistic essentially measures the ratio of within-group variance to the total variance. A smaller Wilks' Lambda indicates greater differences between the groups. In essence, it assesses how much of the total variance in the data is not explained by group membership.

• Hotelling's Trace: This statistic focuses on the sum of the eigenvalues of the between-groups covariance matrix. Eigenvalues represent the magnitude of variance explained by each dimension of the data. A larger Hotelling's Trace suggests greater differences between the groups.

• Pillai's Trace: This statistic is considered more robust than Hotelling's Trace when the assumptions of MANOVA are violated. It measures the sum of the eigenvalues of the between-groups covariance matrix, but each eigenvalue is weighted by the corresponding eigenvalue of the total covariance matrix. This weighting gives more emphasis to the dimensions with larger overall variance.

The choice of which statistic to use can depend on factors like sample size, the number of dependent variables, and the specific research question.

MANOVA Equation

Y = Xβ + ε

• Y: A matrix representing the observed data, where each row corresponds to an observation and each column corresponds to a dependent variable.

• X: A design matrix specifying the group membership for each observation.

• β: A matrix of coefficients representing the group effects on the dependent variables.

• ε: A matrix of error terms, assuming a multivariate normal distribution.

In the context of hypothesis testing, a "race condition" can be metaphorically understood as a situation where the outcome of the test is highly susceptible to subtle, uncontrolled factors that can significantly influence the results.

• Data collection biases

• Violated assumptions

• Researcher bias

• External factors

These "uncontrolled events" can significantly affect the validity of the findings, much like how the outcome of a race depends on the relative timing and performance of the competitors.

To minimize these "race conditions" and ensure reliable results, researchers must carefully design studies, rigorously check assumptions, be mindful of potential biases, and transparently document their methods.

1. Likelihood Ratio Test (LRT)

• (Asymptotic Distribution):

  • -2 * log(Likelihood under H₀) / log(Likelihood under H₁)

2. Bayesian Inference

• Bayes' Theorem):

  • Posterior Probability(H₀ | Data) = [Likelihood(Data | H₀) * Prior Probability(H₀)] / [Evidence]

    • Where:

      • Evidence = Likelihood(Data | H₀) * Prior Probability(H₀) + Likelihood(Data | H₁) * Prior Probability(H₁)

3. Power Analysis

• (For a t-test):

  • Power = P(t > t_critical | H₁ is true)

    • Where:

      • t_critical is the critical t-value for the chosen significance level and degrees of freedom.

      • The calculation of the t-value under the alternative hypothesis depends on the effect size, sample size, and standard deviation.

The Deductive-Nomological Model (D-N Model) of scientific explanation, also known as Hempel's model or the covering law model, proposes that a scientific explanation consists of deducing the phenomenon to be explained (the explanandum) from a set of accepted laws and specific initial conditions (the explanans).

1. The explanandum logically follows from the explanans. This means the explanation must be a valid deductive argument, where the truth of the premises (the explanans) guarantees the truth of the conclusion (the explanandum).

2. The explanans must contain at least one law of nature. These laws are general statements that describe regularities or causal relationships in the natural world. They are crucial for explaining why a particular event occurred.

3. The explanans must be true. The premises used in the explanation must be empirically verified or accepted as true based on existing evidence.

Reference and Citation:

• Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd.

• Neyman, J., & Pearson, E. S. (1933). On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 231(694-706), 289-337.

• Fisher, R. A. (1946). Statistical Methods and Scientific Inference. Oliver and Boyd.

• Student (1908). The Probable Error of a Mean. Biometrika, 6(1), 1-25.

• Pearson, K. (1900). On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to Have Arisen from Random Sampling. Philosophical Magazine, 50(302), 157-175.

• Fisher, R. A. (1925). The Arrangement of Field Experiments. Journal of the Ministry of Agriculture of Great Britain, 33, 503-513.

• Draper, N. R., & Smith, H. (1998). Applied Regression Analysis. Wiley.

• Kaplan, E. L., & Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457-481.

• Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B (Methodological), 34(2), 187-202.

• Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Pearson.