Analysis
Tina Lasisi
April 12, 2023
Last updated: 2023-04-12
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Knit directory: HairManikin2023/
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1 Preparing Data
First, we import the data and label the variables.
1.0.1 Data preview
# Preview data
head(df_wetdry) %>%
kbl(booktabs = T) %>%
kable_styling(latex_options = c("striped", "scale_down"))| wind | wig | wet_dry | heat_loss | skin_temp | resistance | clo | amb_temp | amb_rh | radiation | trial |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.3 | Nude | wet | 90.87576 | 34.01121 | 0.0000000 | 0.0000003 | 34.01091 | 45.81515 | on | 1 |
| 0.3 | Nude | wet | 86.74872 | 34.01205 | -0.0012118 | -0.0078183 | 34.11821 | 45.77179 | on | 2 |
| 1.0 | Nude | wet | 227.27600 | 34.02600 | 0.0001190 | 0.0007674 | 33.99840 | 46.25600 | on | 1 |
| 2.5 | Nude | wet | 276.44615 | 34.02462 | -0.0008197 | -0.0052881 | 34.25051 | 48.20513 | on | 1 |
| 2.5 | Nude | wet | 272.29630 | 34.01741 | -0.0008039 | -0.0051862 | 34.23296 | 48.12963 | on | 2 |
| 0.3 | Straight | wet | 30.00323 | 34.00065 | -0.0042217 | -0.0272367 | 34.12710 | 45.43548 | on | 1 |
1.0.2 Plots
1.1 Removing Outlier
It was noticed that the 2nd trial conducted with wet, tightly curled hair, 2.5 m/s wind speed, and radiation on, had more heat loss than any of the trials with radiation off. With the understanding that radiation should always decrease heat loss, we elected to remove that data point.
# Remove specific entry
df_wetdry <- df_wetdry %>%
filter(!(wig == "Tightly\nCurled" & wind == 2.5 & radiation ==
"on" & wet_dry == "wet" & trial == "1"))
2 Regression models
2.0.1 Radiation off
Here, we model the effect of the wig variable on the
off (heat loss without radiation) variable while correcting
for wind.
Without radiation, having hair will reduce the heat loss.
Call:
lm(formula = off ~ wind + wig, data = df_dry_off)
Residuals:
Min 1Q Median 3Q Max
-8.0303 -3.9809 0.1861 2.6542 14.6310
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 46.156 2.250 20.52 < 2e-16 ***
wind 11.270 1.009 11.17 2.13e-12 ***
wigStraight -40.341 2.618 -15.41 4.46e-16 ***
wigModerately\nCurled -40.747 2.618 -15.56 3.38e-16 ***
wigTightly\nCurled -38.362 2.618 -14.65 1.76e-15 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.555 on 31 degrees of freedom
Multiple R-squared: 0.9384, Adjusted R-squared: 0.9305
F-statistic: 118.1 on 4 and 31 DF, p-value: < 2.2e-162.0.2 Radiation on
With radiation, there is a net increase in heat (i.e. heat gain) without any hair. Additonally, we observe that heat gain decreases with increasingly curled hair.
Call:
lm(formula = on ~ wind + wig, data = df_dry_on)
Residuals:
Min 1Q Median 3Q Max
-13.6776 -4.8542 -0.0306 3.3559 19.1058
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -129.327 2.835 -45.61 < 2e-16 ***
wind 17.406 1.271 13.69 1.1e-14 ***
wigStraight 69.844 3.300 21.16 < 2e-16 ***
wigModerately\nCurled 91.558 3.300 27.74 < 2e-16 ***
wigTightly\nCurled 113.668 3.300 34.44 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 7.001 on 31 degrees of freedom
Multiple R-squared: 0.98, Adjusted R-squared: 0.9775
F-statistic: 380.4 on 4 and 31 DF, p-value: < 2.2e-162.0.3 Solar influx
Here, we model the effect of the wig variable on
influx while correcting for wind.
In the dry heat loss experiments, we see that all hair (regardless of curliness) decreases the solar influx. Additionally, the curlier the hair, the lower the solar influx.
Call:
lm(formula = influx ~ wind + wig, data = df_dry)
Residuals:
Min 1Q Median 3Q Max
-8.079 -3.816 1.087 2.763 9.105
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 175.4829 2.0133 87.161 < 2e-16 ***
wind -6.1369 0.9028 -6.798 1.3e-07 ***
wigStraight -110.1848 2.3434 -47.019 < 2e-16 ***
wigModerately\nCurled -132.3051 2.3434 -56.459 < 2e-16 ***
wigTightly\nCurled -152.0302 2.3434 -64.876 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.971 on 31 degrees of freedom
Multiple R-squared: 0.9939, Adjusted R-squared: 0.9932
F-statistic: 1272 on 4 and 31 DF, p-value: < 2.2e-162.0.4 Summary of dry heat loss regression models
| Radiation Off | Radiation On | Solar Influx | |
|---|---|---|---|
| (Intercept) | 46.16 *** | -129.33 *** | 175.48 *** |
| [41.57, 50.74] | [-135.11, -123.54] | [171.38, 179.59] | |
| wind | 11.27 *** | 17.41 *** | -6.14 *** |
| [9.21, 13.33] | [14.81, 20.00] | [-7.98, -4.30] | |
| wigStraight | -40.34 *** | 69.84 *** | -110.18 *** |
| [-45.68, -35.00] | [63.11, 76.58] | [-114.96, -105.41] | |
| wigModerately Curled | -40.75 *** | 91.56 *** | -132.31 *** |
| [-46.09, -35.41] | [84.83, 98.29] | [-137.08, -127.53] | |
| wigTightly Curled | -38.36 *** | 113.67 *** | -152.03 *** |
| [-43.70, -33.02] | [106.94, 120.40] | [-156.81, -147.25] | |
| N | 36 | 36 | 36 |
| R2 | 0.94 | 0.98 | 0.99 |
| *** p < 0.001; ** p < 0.01; * p < 0.05. | |||
Regression coefficients across regression models.
2.0.5 Dry heat loss ANOVA plots
ANOVA of dry heat loss
| Version | Author | Date |
|---|---|---|
| 2175c5e | Tina Lasisi | 2023-03-25 |
2.1 Evaporative resistance (wet experiments)
Here, we repeat the same modelling process for the evaporative resistance data from the wet experiments.
2.1.1 Radiation off
Here, we model the effect of the wig variable on the
off (heat loss without radiation) variable while correcting
for wind.
Without solar radiation, all hair (regardless of texture) decreases evaporative resistance.
Call:
lm(formula = off ~ wind + wig, data = df_wet_off)
Residuals:
Min 1Q Median 3Q Max
-32.382 -6.006 2.673 5.870 40.839
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 171.043 9.132 18.73 8.71e-13 ***
wind 42.585 3.929 10.84 4.69e-09 ***
wigStraight -116.024 10.179 -11.40 2.20e-09 ***
wigModerately\nCurled -129.170 10.179 -12.69 4.26e-10 ***
wigTightly\nCurled -134.409 10.695 -12.57 4.95e-10 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 16.81 on 17 degrees of freedom
Multiple R-squared: 0.9551, Adjusted R-squared: 0.9446
F-statistic: 90.46 on 4 and 17 DF, p-value: 3.177e-112.1.2 Radiation on
With radiation, hair decreases evaporative resistance.
Call:
lm(formula = on ~ wind + wig, data = df_wet_on)
Residuals:
Min 1Q Median 3Q Max
-47.426 -11.303 4.423 6.822 54.290
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 117.541 12.726 9.236 4.90e-08 ***
wind 55.445 5.475 10.127 1.29e-08 ***
wigStraight -106.632 14.186 -7.517 8.44e-07 ***
wigModerately\nCurled -113.898 14.186 -8.029 3.48e-07 ***
wigTightly\nCurled -123.891 14.905 -8.312 2.16e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 23.42 on 17 degrees of freedom
Multiple R-squared: 0.9255, Adjusted R-squared: 0.908
F-statistic: 52.8 on 4 and 17 DF, p-value: 2.297e-092.1.3 Solar influx
Combining the above data to calculate solar influx, we see that there is not a considerable effect of radiation on evaporative resistance.
Call:
lm(formula = influx ~ wind + wig, data = df_wet)
Residuals:
Min 1Q Median 3Q Max
-13.4512 -4.2205 -0.7951 3.9758 15.0438
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 53.502 4.585 11.669 1.54e-09 ***
wind -12.860 1.973 -6.520 5.24e-06 ***
wigStraight -9.392 5.111 -1.838 0.08368 .
wigModerately\nCurled -15.272 5.111 -2.988 0.00826 **
wigTightly\nCurled -10.518 5.370 -1.959 0.06676 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 8.439 on 17 degrees of freedom
Multiple R-squared: 0.7493, Adjusted R-squared: 0.6903
F-statistic: 12.7 on 4 and 17 DF, p-value: 5.753e-052.1.4 Summary of evaporative heat loss regression models
| Radiation Off | Radiation On | Solar Influx | |
|---|---|---|---|
| (Intercept) | 171.04 *** | 117.54 *** | 53.50 *** |
| [151.78, 190.31] | [90.69, 144.39] | [43.83, 63.18] | |
| wind | 42.58 *** | 55.44 *** | -12.86 *** |
| [34.30, 50.87] | [43.89, 67.00] | [-17.02, -8.70] | |
| wigStraight | -116.02 *** | -106.63 *** | -9.39 |
| [-137.50, -94.55] | [-136.56, -76.70] | [-20.17, 1.39] | |
| wigModerately Curled | -129.17 *** | -113.90 *** | -15.27 ** |
| [-150.65, -107.69] | [-143.83, -83.97] | [-26.06, -4.49] | |
| wigTightly Curled | -134.41 *** | -123.89 *** | -10.52 |
| [-156.97, -111.84] | [-155.34, -92.45] | [-21.85, 0.81] | |
| N | 22 | 22 | 22 |
| R2 | 0.96 | 0.93 | 0.75 |
| *** p < 0.001; ** p < 0.01; * p < 0.05. | |||
Regression coefficients across regression models.
2.1.5 Evaporative heat ANOVA plots
ANOVA of evaporative heat loss
| Version | Author | Date |
|---|---|---|
| 2175c5e | Tina Lasisi | 2023-03-25 |
3 Calculating Thermal Resistance
\[I_t = \frac{T_{Skin} - T_{Air}}{H_{Dry}}\]
df_wetdry["dry_heat_resistance"] <- (df_wetdry["skin_temp"] -
df_wetdry["amb_temp"])/df_wetdry["heat_loss"]
# For the dry data, leave this blank
df_wetdry <- df_wetdry %>%
mutate(dry_heat_resistance = ifelse(wet_dry == "wet",
NaN, dry_heat_resistance))
4 Calculating Net Solar Influx
\[I_{Dry} = H_{Dry} - H_{Dry}^{Solar}\] \[I_{Evap} = H_{Evap} - H_{Evap}^{Solar}\]
# Average all trials with the same characteristics
df_averaged_trials <- df_wetdry %>%
group_by(wig, wind, radiation, wet_dry) %>%
drop_na(heat_loss) %>%
summarise(heat_loss = mean(heat_loss))
# Pivot the dataframe to incldue radiation on and off
# as part of same event
df_radiation_split <- df_averaged_trials %>%
pivot_wider(names_from = c(radiation), values_from = c(heat_loss)) %>%
rename(heat_loss_off = off) %>%
rename(heat_loss_on = on)
# Calculate the net influx
df_net_influx_plots <- df_radiation_split %>%
group_by(wig, wind) %>%
summarise(wet_dry = wet_dry, net_influx = heat_loss_off -
heat_loss_on)
df_net_influx <- df_net_influx_plots %>%
spread(wet_dry, net_influx)
5 Adjusting Heat Losses to 30 Degrees Celsius
5.1 Dry Heat Loss
\[H_{Dry}^{30^\circ C} = \frac{35 -30}{I_t}\]
# Their calculation
df_wetdry["heat_30"] = (35 - 30)/df_wetdry["dry_heat_resistance"]
# What I would expect df_wetdry['heat_30'] =
# (df_wetdry['skin_temp'] - 30) /
# df_wetdry['dry_heat_resistance']
# Recreate the radiation split dataframe to include
# heat_30
df_averaged_trials <- df_wetdry %>%
group_by(wig, wind, radiation, wet_dry) %>%
drop_na(heat_loss) %>%
summarise(heat_loss = mean(heat_loss), heat_30 = mean(heat_30))
df_radiation_split <- df_averaged_trials %>%
pivot_wider(names_from = c(radiation), values_from = c(heat_loss,
heat_30))5.2 Dry and Wet Heat Losses With Solar Radiation
\[H_{Dry}^{30^{\circ} C,\:Solar} = H_{Dry}^{30^{\circ} C} - I_{Dry}\] \[H_{Wet}^{30^{\circ} C,\:Solar} = H_{Evap + Dry}^{30^{\circ} C} = H_{Evap} + I_{Dry} + H_{Dry}^{30^{\circ} C,\:Solar}\]
dry_heat_30 = df_radiation_split[df_radiation_split$wet_dry ==
"dry", ]
heat_evap = df_radiation_split[df_radiation_split$wet_dry ==
"wet", ]
df_adjusted_solar <- data.frame(dry_heat_loss <- dry_heat_30$heat_30_off -
df_net_influx$dry, wind <- dry_heat_30$wind, wig <- dry_heat_30$wig) %>%
rename(dry_heat_loss = "dry_heat_loss....dry_heat_30.heat_30_off...df_net_influx.dry") %>%
rename(wind = "wind....dry_heat_30.wind") %>%
rename(wig = "wig....dry_heat_30.wig")
df_adjusted_solar["wet_heat_loss"] <- +heat_evap$heat_loss_on +
df_net_influx$dry + df_adjusted_solar$dry_heat_loss
df_adjusted_solar_plots <- df_adjusted_solar %>%
pivot_longer(cols = c("dry_heat_loss", "wet_heat_loss"),
names_to = "wet_dry", values_to = "heat_loss")
6 Calculating Evaporative Potential
\[H_{Max}^{30^{\circ} C,\:Solar} = H_{Wet}^{30^{\circ} C,\:Solar} - H_{Dry}^{30^{\circ} C,\:Solar}\]
df_evaporative_potential <- df_adjusted_solar$wet_heat_loss -
df_adjusted_solar$dry_heat_loss7 Calculating Sweat Requirements
\[Sweat_{Max} = \frac{H_{Max}^{30^{\circ} C,\:Solar} * 3600}{2430}\]
\[ IF \; H_{Dry}^{30^{\circ} C,\:Solar} < 0, \; Sweat_{Zero} = -\frac{H_{Dry}^{30^{\circ} C,\:Solar} * 3600}{2430} \\ ELSE, \; Sweat_{Zero} = 0\]
# Create a new df with the sweat requirements
df_sweat_requirements <- data.frame(sweat_max <- df_evaporative_potential *
3600/2430, sweat_zero <- -3600/2430 * df_adjusted_solar["dry_heat_loss"],
wig <- df_adjusted_solar$wig, wind <- df_adjusted_solar$wind)
# Rename columns
colnames(df_sweat_requirements) <- c("sweat_max", "sweat_zero",
"wig", "wind")
# Replace all values less than 0 with 0 per formula
df_sweat_requirements["sweat_zero"][df_sweat_requirements["sweat_zero"] <
0] <- 0
R version 4.2.2 (2022-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Ventura 13.2.1
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] kableExtra_1.3.4 PMCMRplus_1.9.6 ggstatsplot_0.11.0
[4] ggstance_0.3.6 jtools_2.2.1 huxtable_5.5.2
[7] broom.mixed_0.2.9.4 patchwork_1.1.2 gridExtra_2.3
[10] fs_1.5.2 knitr_1.41 lubridate_1.9.2
[13] forcats_1.0.0 stringr_1.5.0 dplyr_1.1.1
[16] purrr_1.0.1 readr_2.1.4 tidyr_1.3.0
[19] tibble_3.2.1 ggplot2_3.4.1 tidyverse_2.0.0
[22] workflowr_1.7.0
loaded via a namespace (and not attached):
[1] colorspace_2.0-3 ggsignif_0.6.4 ellipsis_0.3.2
[4] rprojroot_2.0.3 estimability_1.4.1 parameters_0.20.2
[7] rstudioapi_0.14 farver_2.1.1 listenv_0.8.0
[10] furrr_0.3.1 ggrepel_0.9.3 bit64_4.0.5
[13] fansi_1.0.3 mvtnorm_1.1-3 xml2_1.3.3
[16] codetools_0.2-18 splines_4.2.2 cachem_1.0.6
[19] SuppDists_1.1-9.7 zeallot_0.1.0 jsonlite_1.8.4
[22] broom_1.0.4 Rmpfr_0.9-1 effectsize_0.8.3
[25] compiler_4.2.2 httr_1.4.4 emmeans_1.8.5
[28] backports_1.4.1 assertthat_0.2.1 fastmap_1.1.0
[31] cli_3.6.1 formatR_1.14 later_1.3.0
[34] htmltools_0.5.3 tools_4.2.2 gmp_0.7-1
[37] coda_0.19-4 gtable_0.3.1 glue_1.6.2
[40] Rcpp_1.0.9 jquerylib_0.1.4 vctrs_0.6.1
[43] svglite_2.1.1 nlme_3.1-160 insight_0.19.1
[46] xfun_0.35 globals_0.16.2 ps_1.7.2
[49] rvest_1.0.3 timechange_0.1.1 lifecycle_1.0.3
[52] future_1.29.0 getPass_0.2-2 MASS_7.3-58.1
[55] scales_1.2.1 vroom_1.6.0 ragg_1.2.5
[58] hms_1.1.2 promises_1.2.0.1 parallel_4.2.2
[61] rematch2_2.1.2 prismatic_1.1.1 yaml_2.3.6
[64] memoise_2.0.1 pander_0.6.5 sass_0.4.4
[67] stringi_1.7.8 highr_0.9 paletteer_1.5.0
[70] bayestestR_0.13.0 commonmark_1.9.0 systemfonts_1.0.4
[73] rlang_1.1.0 pkgconfig_2.0.3 evaluate_0.18
[76] lattice_0.20-45 labeling_0.4.2 bit_4.0.5
[79] processx_3.8.0 tidyselect_1.2.0 parallelly_1.32.1
[82] magrittr_2.0.3 R6_2.5.1 generics_0.1.3
[85] multcompView_0.1-8 BWStest_0.2.2 pillar_1.8.1
[88] whisker_0.4 withr_2.5.0 datawizard_0.7.0
[91] performance_0.10.2 crayon_1.5.2 utf8_1.2.2
[94] correlation_0.8.3 tzdb_0.3.0 rmarkdown_2.18
[97] kSamples_1.2-9 grid_4.2.2 callr_3.7.3
[100] git2r_0.30.1 webshot_0.5.4 digest_0.6.30
[103] xtable_1.8-4 httpuv_1.6.6 textshaping_0.3.6
[106] statsExpressions_1.5.0 munsell_0.5.0 viridisLite_0.4.1
[109] bslib_0.4.1