Introduction to mbir

1.1. smd_test()

The objective of smd_test() is to express difference in means between two continuous data vectors (standardized mean difference). The study design may be independent group differences or paired group differences (e.g. pre-post analysis). As the overarching goal of mbir is to provide an automated sequence of statistical decisions, smd_test() descends raw data down the correct path(s) illustrated in Figure 1 and described below:

Figure 1: decision tree illustration

1) Explore Assumptions: In order to perform parametric tests, data must fit parametric assumptions, otherwise the results are likely improper. To start, parametric tests assume a Gaussian (normal) distribution. Rather than visually inspecting normality via histograms or Q-Q plots, quantifying the shape of the distribution allows for objective decision-making. smd_test() employs the Shapiro-Wilk test to check whether the distribution deviates from a comparably normal shape (i.e. compares the scores in the sample to a normally distributed set of scores with the same mean and standard deviation). If the test is non-significant (p < 0.05) informs us that the sample is not significantly different from normal, indicating the assumption is observed. In this case, smd_test() answers “Yes” to the normality assumption and descends left in Figure 1. The next required assumption is homogeneity of variance (equal variance between vectors). This step is quantified via the F-test to examine the ratio of the two variances. As with the Shapiro-Wilk test, non-significance informs us that homogeneity is assumed, and again, answers “Yes” in the decision tree (Figure 1) which ultimately dictates which t-test performed (will be discussed soon).

Before moving on however, if perhaps the data is non-normal, this suggests applying a transformation to counteract the present skewness. Skewness is problematic for t-tests as they are entirely focused on the mean (which is strongly affected by outliers). Unfortunately, deciding whether to remain parametric versus nonparametric is difficult to justify numerically, so smd_test() provides both. On the parametric side, log-transformation is a common technique to correct for positive skewness, and is especially common amongst the sport community.²

• Note: as there is no logarithm of the value zero, if zeros are detected amongst data vectors, a constant of 1 is added to original scores before transformation. The advantage of adding 1 is that the logarithm of 1 is actually equal to zero, so scores of zero before transformation conveniently return to zeros after transformation.

The nonparametric alternative is to rank-transform (i.e. lowest score is given rank 1, next highest score rank 2, etc), which permits less restrictive distributional assumptions than it’s parametric counterpart. The analysis is then carried out on the ranks in lieu of the raw data, which is an ingenious maneuver around the parametric assumptions.

2) Null-Hypothesis Statistical Testing: Once the assumptions and/or transformations are accounted for, the succeeding step is to perform the determined hypothesis test. On the parametric side (after considering normality) the homoscedastic assumption dictates the type of t-test applied. To quickly substantiate, if the two variances are equal there is only one source of uncertainty to account for as the standardizer (developed by WS Gosset “Student”), whereas there are two sources of uncertainty to account for if variances are unequal (BL Welch method). On the nonparametric side, the Wilcoxon derivatives (sign-rank, paired design; rank-sum, independent design) are the nonparametric, rank-transformed equivalent to the t-tests.

3) Effect Size: Upon acquiring NHST parameters, the job is not done. Dogmatic p-values fail to express the level of impact, or magnitude, of effect - as this parameter is confounded on both n and observed effect,³ leading to deceitful interpretations. Rather, a popular solution to this critism is to standardize the magnitude of effect, known as effect size, and thus serves as a foundation of mbir to supplement the determination of effects under consideration. Numerous effect size measures exist, but the most common are Cohen’s d, Pearson’s r and odds ratios, and are one of the final steps of smd_test() to programmatically calculate from the corresponding NHST parameters.

In a nutshell, when a Gaussian distribution is detected, the terminal effect size is Cohen’s d, which is purely the effect size in original units divided by the appropriate standard deviation (which mbir automatically detects). Therefore, Cohen’s d can be interpreted as untis of standard deviations, and is sometimes called standardized mean difference, hence the “smd” acronym.

• Unfortunately, d overestimates size of effects with small sample sizes. Consequently, to reduce bias and avoid overestimating d, an approximate adjustment factor is applied to d when degrees of freedom are below 30.⁴ Thereafter, d_adj represents the adjusted effect size value.

Upon log-transformation, exponentiating the confidence intervals of the log-transformed difference consequently bestows a difference in geometric means.⁵ Subsequently taking the difference between these two values captures the point estimate - that is, on the log scale: log(a)-log(b) = log(a/b), which can be interpreted in a similar manner as an odds ratio. On the nonparametric side, dividing the z value by the square root of n stealthily arrives at r.

• Effect size confidence intervals are provided for each analysis to disclose the precision of observed effect size, or perhaps the maximum likely estimation error.

4) Magnitude-Based Inferences:

The final calculations aggregate the aforementioned statistical parameters to ultimately formulate magnitude-based inferences of effects. Calculations were reverse-engineered from Hopkins’s cited spreadsheet⁶ and transcribed into the R language. Users are welcome to use mbir Input Functions (Section 2) to directly compare their numerical results to the respective spreadsheet and thereby confirm mbir calculations are correct.

To quickly abbreviate, magnitude-based inferences offer a form of inference in which the parameter is partitioned into three regions corresponding to various qualitative aspects (substantially negative, trivial, substantially positive). A probability is assigned to each of these regions, calculated from the assumed distribution of the parameter estimate. To be more transparent, the obtained t statistic is adjusted (updated) by the calculated effect size before justifying the cumulative probability distribution. (This marvelous statistical modulation has therefore been coined “pseudo-Bayesian”). Each mbir function presents the observed magnitude of effect, as well as the precision of estimation. P-values are shown for the sake of formality, but users are encouraged to highlight Hopkins’ content-rich inference: making a qualitative statement with regard to practical significance. The last line of smd_test() output composes the appropriate statement from given parameters, which considers both the level of confidence (possibly, likely, etc), level of magnitude (trivial, small, moderate, etc), as well as the direction (negative, positive) of effect.

Unfortunately magnitude-based inferences has been formerly criticized for inflated probabilities of effect (lacking conservativeness).^{7 , 8} To counter this notion however, if an intervention is able to reproduce a possibly small improvement in elite performance, it is therefore an arguably worthwhile and meaningful intervention (that may have passed under the statistical radar of more conservative approaches).

Example 1
Consider the following data set, which sampled twenty wide receiver 40-yard dash times from both 2007 and 2017 NFL combine.⁹

If a practioner wanted to decipher if the prospective athletes recorded faster times in 2017 versus 2007, smd_test() would follow the below syntax:

• x as the group of interest (2017).
• y as the null group to test against (2007).
• paired = FALSE due to the independent study design.

mbir::smd_test(x = now, y = then, paired = FALSE, auto = T)

##    Mean of now = 4.53; Mean of then = 4.47
##    Normality Observed, Equal Variance
##    Method:  Two Sample t-test
## 
##           Raw Scale
## t             2.095
## df           38.000
## P value       0.043
## 90% CI LL     0.011
## 90% CI UL     0.104
## 
##    Cohen d = 0.66
##    90% CI [0.13, 1.2]
## 
##    Magnitude-Based Inference 
## 
##         Most Unlikely Possibly  Possibly
##         "Lower"       "Trivial" "Higher"
## MBI (%) "0"           "30.5"    "69.5"  
## 
## Inference: Possibly Moderate Increase.

smd_test() output first provides the mean for the respective groups, followed by the detected assumptions and statistical test performed. Since normality and homoscedasticity were observed, the Student’s t-test was executed upon the raw data (which can be confirmed by following the decision tree path in Figure 1). After displaying the t-test parameters, we see Cohen’s d of 0.66 (± 0.54) attributes to a moderate effect size magnitude (according to Hopkins’ thresholds). However, applying the qualitative language associated with the magnitude-based inferences (last line of smd_test() output), we can add possibly to our conclusion statement.

Since the effect size unit ended up positive, this infers that 2017 times were actually slower than 2007 (an increase in time). Therefore, the concluding statement to the above scenario would be something to the effect of:

“Wide receivers from the 2017 NFL combine recorded slower 40-yard dash times than wide receivers from the 2007 NFL combine by a possibly moderate degree (d = 0.66 ± 0.54).”

Example 2

Consider this fictional data set of pre-test post-test pull up values. Notice the large positive skewness, as the majority of pre-test values reside between 0-4 with a handful of outliers who record more than 6 pull ups.

If a practitioner wanted to ask if there was a positive effect after some training intervention, smd_test() would be used in the following manner:

• x post-test
• y pre-test
• paired = TRUE

# Named a variable "mag" for later use within vignette
mag<-mbir::smd_test(post, pre, paired = T, auto = T)

##    Mean of post = 3.08; Mean of pre = 1.96
##    Skewness Observed, Equal Variance
##    Method: Paired t-test
## 
##         Log-Transformed
## t               2.53009
## df             24.00000
## P value         0.01838
## 
##    Factor = 1.53
##    90% CI [1.15, 2.05]
## 
##    Magnitude-Based Inference
##  
##         Most Unlikely Very Unlikely Most Likely
##         "Lower"       "Trivial"     "Higher"   
## MBI (%) "0.2"         "3.6"         "96.2"     
## Inference: Most Likely Small Increase.
## -----------------------------------------------
##    Nonparametric Inference
##    Method: Wilcoxon signed rank test
## 
##           Rank-Based
## V            144.000
## P value        0.046
## 90% CI LL      0.000
## 90% CI UL      2.500
## 
##    r = 0.28
##    90% CI [-0.06, 0.57]
## 
##    Magnitude-Based Inference 
## 
##         Very Unlikely Unlikely  Likely    
##         "Negative"    "Trivial" "Positive"
## MBI (%) "3.4"         "15.2"    "81.4"    
## 
## Inference: Likely Small Effect Size

Since the raw data deviated far from a Gaussian distribution, transformations were applied to correct for such violation. Regarding the parametric inference, quite significant NHST parameters are calcualted. However, what is more telling is that post-test values recorded 1.53 (± 0.45) times more pull ups than pre-test, resulting in a most likely small increase. Before moving on though, let’s consider the philosophy behind Frank Wilcoxon’s underutilized, yet elegant, nonparametric methodology - as it may provide better context for answering the present question.

The Wilcoxon signed-rank test operates in a fairly similar manner as the paired t-test (based on the difference between two conditions). Once these differences between pre-/post-test are calcualted they are numerically ranked with the sign of the difference (positive/negative) assigned to each rank (zeros excluded). In the end, the sum of the positive ranks are tested against the sum of the negative ranks. Thinking back to the pull up scenario, this logic dovetails nicely with the question at hand: how many athletes improved, performed equal, or worse from pre-test to post-test? The nonparametric analysis indeed appreciates that a select few experienced a performance decay (r 90% lower bound: -0.06) while most reaped a positive adaptation (r 90% upper bound: 0.57). Therefore, the present conclusion would take on the following argot:

“After 4-week training intervention, athletes exhibited a likely small increase in pull up performance (r = 0.28 ± 0.32).”

1.2. corr_test()

The objective of corr_test() is to express associations between two continuous data vectors. Paralleling swd_test(), cor_test() accepts raw data and provides an automated sequence of statistical decisions. The statistical progression for correlation is far more superficial compared to difference of means. The major assumption of Pearson correlation is that the data follows a bivariate Gaussian disribution. If the given data set deviates from said distribution, Spearman’s rank correlation is implemented instead (association of rank-transformed data). Spearman’s standardized unit of measurement is ρ (rho), but is interpreted the same as the linear r. Simply supplying two data vectors to corr_test() (x, y) delivers an output analogous to smd_test().

2.1. smd()

The syntactic structure of smd() requires the following t-test parameters:

• effect size measure (Cohen’s d)
• p-value
• degrees of freedom

mbir::smd(es = 0.7, p = 0.02, df = 24)

##    Standardized Mean Difference:
##    es = 0.7
##    p value = 0.02
##    90% CI [0.22, 1.18]
## 
##    Magnitude-Based Inference 
## 
##         Most Unlikely Unlikely  Likely    
##         "Negative"    "Trivial" "Positive"
## MBI (%) "0"           "24.2"    "75.8"    
## 
## Inference: Likely Small Increase.

2.2. corr()

The syntactic structure of corr() requires the following correlation parameters:

• r
• n

mbir::corr(r = 0.22, n = 30)

##    Correlation Coefficient:
##    r = 0.22
##    n = 30
##    90% CI [-0.09, 0.49]
## 
##    Magnitude-Based Inference 
## 
##         Very Unlikely Unlikely  Possibly  
##         "Negative"    "Trivial" "Positive"
## MBI (%) "4.6"         "21.5"    "73.9"    
## 
## Inference: Possibly Small Positive Correlation.

2.3. odds()

The syntactic structure of odds() requires the following test parameters:

• odds ratio
• p-value

mbir::odds(or = 3.25, p = 0.06)

##    Odds Ratio:
##    or = 3.25
##    p value = 0.06
##    90% CI [1.16, 9.11]
## 
##    Magnitude-Based Inference 
## 
##         Very Unlikely Very Unlikely Most Likely
##         "Negative"    "Trivial"     "Positive" 
## MBI (%) "2"           "2.3"         "95.7"     
## 
## Inference: Most Likely Small Increase

2.4. prop()

A moderately common research question asks about the relation, or difference, between two nominal variables, producing a 2x2 data set called a frequency table. The approach in answering questions about such a table is to calcualte the difference between two proportions.

The syntactic structure of prop() simply requires the proportions and sample sizes of each group under consideration:

mbir::prop(p1, n1, p2, n2)

For a brief example, consider this fictitious plot:

Imagine there were two teams that recorded the same number of minor overuse injuries during the off-season, say 10 athletes. However, someone recognized that the roster size for Team 1 was 19, whereas the roster size for Team 2 was 26 - that is, on a relative scale, over half of Team 1 accrued an overuse injury (10/19 = 53%) versus only 38% of Team 2 (10/26). Upon immediate realization, 53% was perceived as substantially higher than 38%, causing some people to question whether the coach of Team 1 should be castigated. However, before jumping to rash conclusions, this person first utilized prop() to help construct the proper locution. Let’s peer inside the results of the given scenerio:

mbir::prop(p1 = 0.53, n1 = 19, p2 = 0.38, n2 = 26)

##    Test of Two Proportions:
##    diff = -0.15
##    chi-sq = 1
##    90% CI [-0.37, 0.09]
##    p value = 0.36
## 
##    phi = 0.15
## 
##    Magnitude-Based Inference 
## 
##         Possibly  Possibly     
##         "Trivial" "Substantial"
## MBI (%) "37.2"    "62.8"       
## 
## Inference: Unclear Effect Size.

It appears that coach of Team 1 lucked out! There was an unclear differentiation between the two relative measures (also, notice how zero resides between the confidence intervals). However, this was a close call considering that Φ is beyond the small threshold. Perhaps with a slightly larger n magnitude-based inferences would have quantified a possible effect.

• Note: since χ2 distribution is positive, magnitude-based inferences must surpass possibly substantial threshold in order to detect effects. Also, trivial effects must not transcend unlikely threshold.

2.5. corr_diff()

Upon testing correlations via corr_test() or corr(), users may want to decipher a post hoc difference between two acquired correlations. The syntactic structure of corr_diff() requires the following test parameters:

mbir::corr_diff(r1, n1, r2, n2)

For a brief example, consider this fictitious plot:

Suppose a practitioner used corr_test() twice to derive correlations between subjective body image satisfaction scores and overall well-being for both female and male athletes. The female group exhibited a most likely large positive association of r = 0.53 (n = 25), while the male group revealed a most likely moderate positive association of r = 0.41 (n = 38). However, if the ultimate question was whether body satisfaction is more essential to well-being for females than males, the practitioner needs to test the confidence interval on the difference:

# Female correlation test
female<-mbir::corr_test(F$bodyimage, F$wellbeing)

# Male correlation test
male<-mbir::corr_test(M$bodyimage, M$wellbeing)

# Test of Two Correlations
mbir::corr_diff(r1 = male$r.stat, n1 = 38, r2 = female$r.stat, n2 = 30)

##    Test of Two Correlations:
##    diff = 0.12
##    90% CI [-0.28, 0.57]
##    p value = 0.55

Even though the two correlations display strong evidence towards their associations independently, the practitioner cannot conclude that one of the groups has a stronger relationship compared to the other (i.e. the two positive correlations do not differ). In the end, the conclusion would be there is considerable uncertainty between groups, as the difference in correlations for females and males (r₂ - r₁ = 0.12) could plausibly be as low as -0.28 or as high as 0.57.

3.1. ss_smd()

The syntactic structure of ss_smd() requires the following parameters:

• exp desired size of experimental group
• con desired size of control group
• es desired effect size (Cohen’s d)

mbir::ss_smd(exp = 20, con = 18, es = 0.6)

##    n = 38
##    es = 0.6
## 
##    Estimated:
##    Magnitude-Based Inference 
## 
##         Very Unlikely Unlikely  Likely    
##         "Negative"    "Trivial" "Positive"
## MBI (%) "0.9"         "10.4"    "88.7"    
## 
## Inference: Likely Moderate Increase.

3.2. ss_corr()

The syntactic structure of ss_corr() requires the following parameters:

• n desired number of data pairs
• r desired correlation coefficient

mbir::ss_corr(n = 25, r = 0.25)

##    n: 25
##    r: 0.25
## 
##    Estimated:
##    Magnitude-Based Inference 
## 
##         Very Unlikely Unlikely  Likely    
##         "Negative"    "Trivial" "Positive"
## MBI (%) "4.8"         "18.5"    "76.7"    
## 
## Inference: Likely Small Positive Correlation.

3.3. ss_odds()

The syntactic structure of ss_odds() requires the following parameters:

• exp desired size of experimental group
• con desired size of control group
• odds desired odds ratio

mbir::ss_odds(exp = 12, con = 16, or = 2.5)

##    n = 28
##    OR = 2.5
## 
##    Magnitude-Based Inference 
## 
##         Unlikely   Unlikely  Likely    
##         "Negative" "Trivial" "Positive"
## MBI (%) "8.9"      "5.3"     "85.8"    
## 
## Inference: Unclear Difference.

3.4. aipe_smd()

As mentioned above, AIPE in centered around the idea of precision. How wide could a confidence interval be while still providing useful information? What is the maximum error of estimation we are willing to accept? These are the forms of questions that can be answered via AIPE. The unit of error estimation for AIPE is the margin of error (MOE), expressed in standard deviation units, f. Luckily, Cohen’s d is also interpreted as a fraction of standard deviations. Therefore, when working with aipe_smd(), feel free to think of MOE synonymous with Cohen’s d.

Suppose a team of investigators establish an independent study design with the desired precision of f = 0.5 at the 90% confidence level. (That is, with a 90% CI they are willing to accept a Cohen’s d of ± 0.5). Let’s examine how this team would implement aipe_smd():

• moe 0.5 target margin of error
• paired = F independent design
• conf.int (optional) defaults to 90%

mbir::aipe_smd(moe = .5, paired = F)

## Independent Design       
## MOE 0.5
## CI  0.9
## 
## Estimated Sample Size (per group)
## n = 24
## 
## With 99% Assurance
## n = 34

The output demonstrates that in order to achieve MOE no more than 0.5 the team needs at least 24 subjects in each group. Taking this one step further, if the team wanted to know what percent of experiments would produce a MOE smaller than their target MOE, they would refer to the second given sample size regarding assurance. (Assurance calculates n based upon how many experiments would yield more precision than the target MOE in the long-run). To interpret the present scenario, 99 out of 100 experiments with n = 34 per group would provide more precision than 0.5.

• aipe_smd() defaults to 99% assurance, but users are welcome to adjust with assur.lvl arugment.

Shifting to the paired design, the only additional argument is to provide the correlation between the two measures. At first, pre-test post-test correlations may seem a bit tricky to project, however, referring to prior literature gives good orientation of what previous researchers have observed. r is also quite influential in AIPE calculations, so please be mindful when establishing r.

An example paired design would therefore follow the structure of:

mbir::aipe_smd(moe = .5, paired = T, r = .65)

## Paired Design        
## MOE 0.50
## CI  0.90
## r   0.65
## 
## Estimated Total Sample Size
## n = 10
## 
## With 99% Assurance
## n = 16

Due to the underlying sensitivity paired designs present, notice the considerable reduction in sample size required to obtain equal precision.

4.1. es_convert()

If perhaps users are not familiar with a given effect size measure from say, smd_test(), they are encouraged to convert between equivalent effect size unit of choice. es_convert() provides the service of converting between Cohen’s d, r, and odds ratio.^{14 , 15}

Example

The syntactic structure of es_convert() is of the form:

mbir::es_convert(x, from = c("d", "or", "r"), to = c("d", "or", "r"))

Simply entering the final effect size measure (say, a pearson r of 0.20), declaring the current unit (from = "r"), and stating what unit to convert to (to = "or") delivers:

mbir::es_convert(0.2, from = "r", to = "or")

## Effect Size Conversion
## r = 0.2 
## or = 2.1

If users wish to include effect size confidence intervals from preceding smd_test() results within one line of code, they may do so as such:

# Section 1.1. pre-test post-test 40 yard dash example:
mag<-mbir::smd_test(post, pre, paired = T, auto = T)

# Wish to convert given Odds Ratio to Cohen's d:
mbir::es_convert(c(mag$or.LL, mag$or.stat, mag$or.UL), from = "or", to = "d")

## Effect Size Conversion
## or = 1.15 1.53 2.05
## d = 0.08 0.23 0.4

4.2 boot_test()

Perhaps a given data set does not follow a Gaussian distribution, or even after log-transformation there are still major outliers present. If this is the case, bootstrapping is an attractive alternative for deriving nonparametric confidence intervals via resampling, as it does not assume a distribution. The simple, yet exquisite, bootstrap procedure is as follows:

1) Sample n observations (with replacement) from each independent data vector.

2) Compute the statistic of interest from each “bootstrapped” sample (mbir calculates the median to ensure a robust estimation of central tendency).

3) Calculate and record the difference in medians between the two “bootstrapped” samples.

4) Repeat the above steps a large number of times to obtain a list of difference in medians (mbir defaults to 10,000 repetitions).

5) Finally, the bootstrap confidence interval is simply the interval that contains the lower and upper bound of the observed difference in medians. mbir defaults to 90% confidence interval, so in this case, the 5th and 95th percentile of this final vector is the 90% bootstrap confidence interval.

• The method described above is called percentile-based resampling confidence interval. Subsequent versions may entail other bootstrap derivatives upon user requests.

Example

Consider a data set representing sRPE scores from both “low” and “high” days, respectively. Let’s say a user wants to determine if they can definitively say that their athlete’s “high” sessions are being perceived higher than the “low” sessions. After noticing the apparent skewness presented in “low” data set, the user could use boot_test() as a hasty resource to evade any distributional assumptions (smd_test() would also be appropriate, of course, but boot_test() will be used for sake of example).

The syntactic structure of boot_test() is of the form:

mbir::boot_test(x, y, conf.int, resample, med)

The only required arguments are to provide two numeric data vectors for x and y. Beyond this, specified confidence intervals may be supplied (default is 90%), number of resampling repititions (defaults to 10,000), as well as a specified true median to test against (defaults to 0).

In our example, the user specifically wants to test whether “high” sessions are at least 100 units above “low” sessions in order to definitively say there is a difference. Therefore, boot_test() would be used in the following manner:

mbir::boot_test(high, low, med = 100)

## Median of high = 420; Median of low = 276
## 
## 90% Bootstrap Confidence Interval
## 10,000 Resamples
## 
##  5% 95% 
##  69 224 
## 
## Difference in Medians: Lacking Evidence (CI contains 100).

The histogram illustrates the bootstrapped distribution of the resampled difference in medians. If the dashed line (med) is within the solid lines (confidence intervals), evidence of effect is absent. Conversely, if med is outside the confidence intervals, evidence of effect is present.

Regarding the sRPE data set, surely an effect is present, but unfortunately the user can not definitively say there is a difference of 100 after resampling, as the 90% confidence interval contains 100 (dashed line between solid lines).

4.3 swc_ind()

Applying magnitude-based inferences to the longitudinal trend of an athlete’s performance measurements can be implemented with swc_ind(). Two types of analyses can be explored with swc_ind(): magnitude of change from an individual’s preceding measurement, and the degree of deviation from their overall trend.¹⁶

Specifying “previous” under the type argument will execute the first type of analysis, illustrating magnitude-based inferences regarding the change between each time point. Simply supply raw measurement values in ascending chronological order (first measurement in row 1, most recent measurement in nth row) and swc_ind() will provide partitioned magnitude-based inferences accompanied with qualitative inferences of either increase or decrease from previous time point. The function also provides a visual demonstrating relative change, including vertical error bars (typical error) with each data value. Measurements are linearly connected to express the inherent repeated measure analysis, while underlaying the smallest worthwhile interval bands to supplement the visual trend.

For “previous” analysis, the syntactic structure of swc_ind() involves the following parameters:

• x numeric data vector of an individual’s performance measurements
• swc (±) smallest worthwhile change
• type “previous”
• te (optional) typical error of measurement

df<-c(97.5,99.9,100.2,101.1,101.3,99.8)
mbir::swc_ind(x = df, swc = 0.25, type = "previous")

## MBI From Previous Time Point:
##
##         Diff    N  T  P               MBI
## 1 - 2    2.4    8  4 88   Likely Increase
## 2 - 3    0.3   37 12 51               ---
## 3 - 4    0.9   26  9 65               ---
## 4 - 5    0.2   40 11 49               ---
## 5 - 6   -1.5   76  7 17               ---

Smallest worthwhile change argument requires users expertise regarding their particular data/athlete. With a lack of context, it would negligent to provide a default value for each situation. On the other hand, typical error defaults to the standard error of the linear regression residuals, but practitioners can certainly modify as needed.

Regarding the output table, from left to right, first describes the raw difference from preceding time point in ascending chronololgical order, followed by the probabilistic magnitude-based inference relating to the direction of change (“N” negative, decrease; “T” trivial; “P” positive, increase). On the far right are the qualitative inferences if detected (dashed lines represent unclear change, otherwise specified). The current example displays a likely increase from measurement 1 to measurement 2. Otherwise, no meaningful change was detected beyond time point 2.

Transitioning to the other offered analysis, “trend”, permits user to assess linear trajectory of measurements, as well as the degree of which each measurement deviates from the line of best fit. When “trend” is specified, users will be granted information regarding traditional regression statistics comparing values to a null model (in case of literature pursuit), such as: β coefficient (slope), R², F-ratio and p-value. However, to apply magnitude-based inferences towards the linear trend requires an additional argument to specify; an anticipated slope (±) to test against.

For “trend” analysis, the syntactic structure of swc_ind() involves the following parameters:

• x numeric data vector of an individual’s performance measurements
• swc (±) smallest worthwhile change
• type “trend”
• ts (±) target slope

mbir::swc_ind(x = df, swc = 0.25, ts = 0.2, type = "trend")

##    Trend Parameters:               
## Slope     0.474
## R-squared 0.283
## F stat    2.973
## P value   0.160
##
##         Very Unlikely Unlikely  Likely    
##         "Decrease"    "Trivial" "Increase"
## MBI (%) "3"           "20"      "77"      
##
##    MBI From Trend Line:
##
##  Point   Diff    N  T  P                  MBI
##      1   -1.3   88  7  5         Likely Lower
##      2    0.6   26 12 62                  ---
##      3    0.5   30 13 57                  ---
##      4    0.9   21 11 68                  ---
##      5    0.6   26 13 61                  ---
##      6   -1.4   90  5  5    Very Likely Lower

Below the given Trend Parameters is the magnitude-based inference about the slope against the target slope. In the current example, the slope of 0.474 is likely higher than the specified 0.25. The last table describes magnitude-based inferences of how far each time point vertically deviates from regression line (essentially an analysis of residuals). The current example displays that the first measurement is likely lower than the trend line, as well as measurement 6 being possibly lower. Otherwise, no meaningful deviations were detected.

References