Hypothesis Testing and Descriptive Statistics Formulas

Hypothesis Testing

• Hypothesis tests are statistical methods used to make decisions or inferences about population parameters based on sample data.

Descriptive Statistics Reminders

• These formulas provide ways to compute key measures like the mean, variance, and standard deviation for both ungrouped and grouped data.

Mean (Average)

• For a series of values: x̄ = (∑xi) / N, where ∑xi is the sum of all values and N is the number of values.
• For grouped data: x̄ = (∑xini) / N, where xi is the value, ni is the frequency, and N is the total frequency.

Variance

• For a population: σ² = ∑(xi – x̄)² / N, measures the spread around the population mean.
• For a sample: S² = ∑(xi – x̄)² / (N-1), estimates the population variance using sample data. It includes a correction (N-1) for unbiased estimation.
• SCE: var = SCE/ddl; calculation isn't clearly defined in this context.

Standard Deviation

• Standard Deviation S = √var: Measures the typical deviation of values from the mean, essentially the square root of the variance.

Student's t-Test

• t-Tests are a class of hypothesis tests used to compare means. The appropriate test depends on the nature of the data (paired, independent) and knowledge about variances.

Comparing a Sample Mean to a Theoretical Value

• Use this when you want to determine if the mean of a sample is statistically different from a known or hypothesized population mean (µ0).

Variance of Population Known

• H0: µ1 = µ0 (Null hypothesis: no significant difference).
• H1: µ1 ≠ µ0 (Alternative hypothesis: there is a significant difference).
• Test statistic: zob = |x̄ − µ0| / (σ / √n)
• Decision:
  • If zob < z théo: Reject H1, and cannot reject H0 (Fail to reject null hypothesis) (no significant difference)
  • If zob > z théo: Reject H0 (significant difference)

Variance of Population Unknown

• H0: µ1 = µ0 (Null hypothesis: no significant difference).
• H1: µ1 ≠ µ0 (Alternative hypothesis: there is a significant difference).
• Test statistic: tob = |x̄ − µ0| / (S / √n)
• Decision:
  • If tob < t théo : Fail to reject null hypothesis (Ho retained) (no significant difference)
  • If tob > t théo : Reject null hypothesis H0 (significant difference)

Comparing Two Independent Sample Means

• This is used to see if the means of two independent groups are significantly different.

Variances of Populations Known

• H0: µ1 = µ2 (Null hypothesis: no significant difference between means).
• H1: µ1 ≠ µ2 (Alternative hypothesis: means are different).
• Test statistic: zcal = (x̄1 - x̄2) / √(σ1²/n1 + σ2²/n2)
• Decision:
  • If zcal < z théo: Fail to reject null hypothesis (Ho retained) (no significant difference))
  • If zcal > z théo: Reject null hypothesis H0 (significant difference)

Variances of Populations Unknown

• H0: µ1 = µ2 (Null hypothesis: no significant difference between means).
• H1: µ1 ≠ µ2 (Alternative hypothesis: means are different).
• Test statistic: tcal = (x̄1 - x̄2) / √(Sc²(1/n1 + 1/n2))
  • Sc² = (S1²(n1 - 1) + S2²(n2 – 1)) / (n1 + n2 - 2) (pooled variance estimate)
• Decision:
  • If tcal < t théo: Fail to reject null hypothesis Ho (no significant difference)
  • If tcal > t théo: Reject null hypothesis H0 (significant difference)

Comparing Two Dependent Sample Means (Paired)

• Used when comparing means of two related samples (e.g., before and after treatment on the same subjects).
• H0: µdiff = 0 (Null hypothesis: no average difference between paired observations).
• H1: µdiff ≠ 0 (Alternative hypothesis: there is a difference).
• Test statistic: tcal = x̄diff / (Sdiff / √n)
• Decision:
  • If tcal < t théo: Fail to reject null hypothesis Ho (no significant difference).
  • If tcal > t théo: Reject null hypothesis H0 (significant difference).

Conditions for Applying the t-Test

Normality

• Data should be approximately normally distributed
  • Assess visually (histograms, Q-Q plots).
  • Shapiro-Wilk test for normality.

Homogeneity of Variance (Homoscedasticity)

• For independent two-sample t-tests, the variances of the two groups should be approximately equal.
  • Test using Fisher-Snedecor (F-test)
  • H0: σ1² = σ2² (variances are equal).
  • F = Variance max / Variance min
• Conclusion:
  • If variances are equal: Use the regular Student's t-test.
  • If variances are unequal: Use the modified Student's t-test (Welch's t-test) to correct for unequal variances.

ANOVA (Analysis of Variance) Tests

• ANOVAs are used for comparing means of two or more groups.

Conditions for Performing ANOVA

• Normality: The data within each group should be approximately normally distributed.
• Homogeneity of variances: Variances should be approximately equal across all groups.
  • Use the F test or Bartlett's test (latter for more than two groups). Bartlett is more sensitive to departures from normality
  • F test is used to verify homogeneity of variances.

One-Way ANOVA (ANOVA 1 Facteur)

• Used to compare the means of several groups based on a single factor (independent variable).
• Hypothesis:
  • H0: m1 = m2 = m3 (Null hypothesis: no significant difference between the means of the groups).

ANOVA Table Structure

• Factorial (Between-groups) Variation:
  • SCE = ni * ∑(mi – M)², where ni is the number of individuals, mi is the mean of each group, M is the overall mean, and C is the number of groups.
  • ddl = C-1 (degrees of freedom)
  • CM = ni * ∑(mi – M)² / (C-1)
• Residual (Within-groups) Variation:
  • SCE = SCEtotal – SCEfactorielle
  • ddl = N-C (degrees of freedom)
  • CM = SCE résiduelle / (N-C)
• Total Variation:
  • SCE = ∑(xi – M)²
  • ddl = N-1 (degrees of freedom)
• F-statistic: F = CM Fact / CM Res
• Decision:
  • If Fcal < F tab: Fail to reject null hypothesis H0 ("Pas de différence significative").
  • If Fcal > F tab: Reject null hypothesis H0 ("Différence significative").

Post-Hoc Comparisons (Tukey's HSD)

• Used after a significant ANOVA result to determine which specific groups differ significantly from each other.
  • Examples: m1 ≠ m2 ≠ m3, m1 = m2 ≠ m3, m1 ≠ m2 = m3

Two-Way ANOVA (ANOVA 2 Facteurs)

• Used to examine the effects of two factors (independent variables) on a dependent variable.
• Hypotheses:
  • H0: μ1 = μ2 = µ3 (The first factor has no significant effect on the dependent variable).
  • H0: μ'1 = μ'2 (The second factor has no significant effect on the dependent variable).
  • H0: Interaction: The interaction between the two factors has no significant effect.

ANOVA Table

• Total: Variation in entire data set
• Cellular: Represents variation considering all individual cells
• Factor A: Effect of first factor
• Factor B: Effect of second factor
• Interaction A/B: Interaction between factors
• Residual: Error within groups
• Decision-Making: Analogous to One-Way ANOVA: compare Fcal to Ftab for main effects and interaction.
• Post-hoc Comparisons (Tukey's HSD)
  • If H0 is rejected, this identifies which specific groups are different in the test.

Linear Regression

• Linear regression models the linear relationship between a dependent variable 'y' and one or more independent variables 'x'.

Conditions for Application

• Verify the normality of the observations (y).
• Error on x must be negligible.
• Check data for homoscedasticity.

Equation of the Model

• y = ax + b + ε
  • a : slope
  • b : y-intercept
  • ε : Residual error

Calculating a and b

• a (Slope):
  • a = S²xy / S²x or a = cov_xy / S²x
• Ordonnée à l'origine:
  • b = ȳ – a * x̄

Coefficient of Correlation

• Expressed as * r *, indicates the strength and direction of a linear relationship.
• Ranges from -1 to +1.

Evaluating Regression Quality

• Decomposition of Variance: ∑(yi – ȳ)² = ∑(ŷ – ȳ)² + ∑(yi – ŷ)²
• Coefficient of Determination: r² = (SCEregression / SCEtotal) * 100; Indicates the proportion of variance in 'y' explained by 'x'.
• Test of the Slope: Used to test if there is a significant linear relationship.
  • H0: β = 0 → the variable (y) is linearly independent of (x)
  • Tableau ANOVA: Used to test the slope

ANOVA Table for Regression

• Regression, Residual and Total variations are calculated and assessed via the F statistic.
• Decision:
  • P > 0.05 means fail to reject H0: the variable (y) is linearly independent of (x)
  • P < 0.05 means reject H0: the variable (y) is linearly dependent of (x)

Multiple Linear Regression

• Multiple linear regression extends simple linear regression to include multiple independent variables.
• Equation of the model
• y = a1x1 + a2x2 + a3x + … + apxp + b
• Calculation:
• y = f(x1,x2) = a1x1 + a2x2 + b
• Evaluation:
• Coefficient of determination= r² = (SCE regression / SCE total) * 100
• Test of the slope:
• Table ANOVA is used to define this.
• Decision
• P > 0.05 means fail to reject H0: the variable (y) is linearly independent of (x)
• P < 0.05 means reject H0: the variable (y) is linearly dependent of (x1, x2), in less than one of the points is responsible: β1 ≠ β2 ≠ 0 , β1 = β2 ≠ 0 , β1 ≠ β2 = 0

Non-Linear Regression

• Linearitazion provides a method for computing linear regression on non-linear data.
• Equation Expo= y = beᵃx
• Linearization and Equation of the model =
• Ln y = Ln beaх
• Ln y = Ln b + ax
• Y = B + ax
• Calculate a and B using the equations in the linear regression table.
• NOTE substitute Y for y in the equations and calculate using X = x
• Inverse Model= y = ax / b+x
• Linearization and Equation of the model
  • 1/y = b+x / ax
  • 1/y = b/ax + x/ax
  • 1/y = b/a * 1/x + 1/a
  • Y = AX + B
  • Calculate A and B using the equations in the linear regression table.
  • NOTE substitute Y for 1/y and X for 1/x in the equations and calculate.
• Michaelis Mentan Model V = Vmax [S] / Km + [S]
• Use:
• 1/V= a . 1/[S] + b
• Y = AX + B
• Calculate the same as above.