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--- sift:statistical_parametric_mapping:using_statistical_parametric_mapping_in_biomechanics [2024/12/17 15:54] – [The Math behind SPM] wikisysop
+++ sift:statistical_parametric_mapping:using_statistical_parametric_mapping_in_biomechanics [2024/12/17 20:13] (current) – [Visualizing SPM Results] wikisysop
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 ===== The Math behind SPM =====
+All of the math behind SPM is done internally with Sift, but we give a brief summary of it here for your purposes in Sift. Please refer to the references for more detailed explanations.
+==== The GLM ====
 The basis of SPM begins with modeling our data with a General Linear Model (GLM). A GLM is simply relating our data to an experimental design. This experimental design (in the form of a matrix) represents the experiment, and may represent what group or condition a trial belongs to (i.e. a 1 in the column that the trial belongs to, and 0 otherwise, etc.). We relate these though the formula:
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 ==== T-Tests ====
-For a t-test, the nth term in the SPM is:
+For a t-test, we model the GLM as above, with the design matrix consisting of 1's indicating a grouping for each test, and 0's otherwise. The SPM is evaluated with:
-{{:SPM_TTestEqn.png}}
+{{:spm_ttest_eqn.png}}
-where c is a contrast vector indicating how we are selecting from our regression matrix (B), ^T represents a transposed matrix, rho is the square-root of our variance, and X is our design matrix. The calculation for rho is shown below:
+where c is a contrast vector indicating how we are selecting from our regression matrix (B), ^T represents a transposed matrix, sigma is the square-root of our variance, and X is our design matrix. The calculation for sigma is shown below:
-{{:SPM_TTestEqnRho.png}}
+{{:spm_sigma.png}}
-where e is the residuals, nn represents the nth diagonal element of the matrix, I represents the number of trials we have instituted, and rank(X) is equal to the number of groups (for a t-test, this would be 2).
+where diag() is the diagonal values in a vector, e is the residuals, I represents the number of trials we have modeled (the # of rows in our Y Matrix), and rank(X) is equal to the number of groups (for a t-test, this would be 2).
+This equation for T follows a T-distribution of degrees of freedom equal to I - rank(X).
 ==== ANOVA ====
+For our ANOVA tests, we model the GLM slightly differently. Instead of the design matrix including just the groupings, we add an additional term to represent the group-level effect (i.e. effect of all groups together), which is represented by all 1's in the design matrix. This is because the ANOVA test requires a "full" model (with group-level and individual group effect included), and a "reduced" model(with just the group-level effect modelled), which it compares. The contrast is applied slightly differently as well: we use it to help us form the full model and reduced models, using a projection matrix.
+{{:spm_anova_eqn.png}}
+Where B is the ANOVA regression matrix, X is the design matrix, M is our projection matrix, and R is the "residual forming matrix", which is similar to the projection matrix, and is the matrix which when applied to Y returns the given residuals.
+The SPM follows a F-distribution with degrees of freedom rank(X) and I - rank(X).
+==== Random Field Theory ====
 With n samples in our map, it would be sound to estimate the significance with a Bonferroni correction, but we know that spatially similar data points in biomechanics are intrinsically dependent on each other, and thus the Bonferroni assumption of independence between data points would result in a far more conservative than necessary significance. As such, Random Field Theory (rft) is employed to estimate the dependence between data points (called smoothness), and to establish a threshold for statistical significance of the data. This can be represented with the familiar p-values for simple t-tests.
+===== Which test to use =====
+Choosing your experimental hypothesis is very important, and this should influence the statistical test being undertaken. ANOVA provides us a broad look at all of our data: with the hypothesis that all groups have the same mean, we can easily test IF there is 1 or more groups not following this hypothesis, but we cannot discern which one it is. T-tests on the other hand can specifically tell us if any 2 groups are different, and specifically identify which tests are different.
+For many groups, it is recommended to first use an ANOVA test, and if there is statistical differences, to use post-hoc t-tests with a bonferroni correction (or the Holm–Bonferroni method) to identify which group this is.
+For related groups, it is recommended to use a paired t-test over a two-sample t-test, as it has strictly higher statistical power.
 ===== Visualizing SPM Results =====