ETC5521: Exploratory Data Analysis

.info-box.w-50.bg-white[
These slides are viewed best by Chrome or Firefox and occasionally need to be refreshed if elements did not load properly. See <a href=lecture-07B.pdf>here for the PDF <i class="fas fa-file-pdf"></i></a>. 
]

<br>

---

# .monash-blue[ETC5521: Exploratory Data Analysis]

<br>

<h2 style="font-weight:900!important;">Making comparisons between groups and strata</h2>

.bottom_abs.width100[

Lecturer: *Di Cook*

<i class="fas fa-envelope"></i>  ETC5521.Clayton-x@monash.edu

<i class="fas fa-calendar-alt"></i> Week 7 - Session 2

<br>

]

</style>

---

# Modelling and testing for comparisons

---

# Revisiting .orange[Case study] .circle.bg-orange.white[1] Pest resistance maize

]
.panel[.panel-name[data]
.h200.scroll-sign.f5[

```r
data(gathmann.bt, package = "agridat")
df1 <- gathmann.bt %>%
  pivot_longer(-gen,
    values_to = "abundance",
    names_to = "species"
  ) %>%
  mutate(species = case_when(
    species == "thysan" ~ "Thrip",
    TRUE ~ "Spider"
  ))
skimr::skim(df1)
```

```
## ── Data Summary ────────────────────────
##                            Values
## Name                       df1   
## Number of rows             32    
## Number of columns          3     
## _______________________          
## Column type frequency:           
##   character                1     
##   factor                   1     
##   numeric                  1     
## ________________________         
## Group variables            None  
## 
## ── Variable type: character ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate min max empty n_unique whitespace
## 1 species               0             1   5   6     0        2          0
## 
## ── Variable type: factor ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate ordered n_unique top_counts     
## 1 gen                   0             1 FALSE          2 Bt: 16, ISO: 16
## 
## ── Variable type: numeric ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate mean   sd p0 p25 p50  p75 p100 hist 
## 1 abundance             0             1 6.01 6.36  0 0.6 2.5 10.7 18.4 ▇▂▃▁▂
```
]]
.panel[.panel-name[R]
.f5[

```r
ggplot(df1, aes(gen, abundance, color = gen)) +
  ggbeeswarm::geom_quasirandom(size = 3) +
  facet_wrap(~species, scales = "free") +
  scale_color_discrete_qualitative() +
  guides(color = FALSE) +
  labs(x = "", y = "Abundance") +
  theme(axis.text.x = element_text(
    angle = 90,
    vjust = 0.5,
    hjust = 1
  ))
```
]]

]

]
.w-50[
* The experiment compared abundance of spiders and thrips on *Bt* variety to the abundance of those on isogenic control variety.
{{content}}
]

]

--
* Would you say that the abundance of spiders and/or thrips are comparable between *Bt* variety and isogenic variety?

---

# Two-sample parametric tests: `$t$`-test (thrips)

.flex[
.w-50.br[
* Assumes the two samples are independent and from the `$N(\mu_x, \sigma^2_x)$` and `$N(\mu_y, \sigma^2_y)$`, respectively.

`$$H_0: \mu_x - \mu_y = 0~\text{ vs. }~H_1: \mu_x - \mu_y \neq 0$$`

`$$T^* = \frac{\bar{X} - \bar{Y}}{SE(\bar{X} - \bar{Y})}.$$`
* Assuming `$\sigma^2_x = \sigma^2_y$`, then `$T^* \sim t_{n_x + n_y - 2}.$`
* A `$100(1 - \alpha)\%$` confidence interval for `$\mu_x - \mu_y$` is given as `$(L, U)$` such that:
`$$P(L<\mu_x - \mu_y < U) = 1 - \frac{\alpha}{2}.$$`
* If `$0 \in (L, U)$`, consistent with `$H_0$`.

]
.w-50[
.panelset[
.panel[.panel-name[thrip]
.f5[

```r
with(
  gathmann.bt,
  t.test(thysan[gen == "ISO"],
    thysan[gen == "Bt"],
    alternative = "two.sided",
    var.equal = TRUE,
    conf.level = 0.95
  )
)
```

```
## 
## 	Two Sample t-test
## 
## data:  thysan[gen == "ISO"] and thysan[gen == "Bt"]
## t = -3.2182, df = 14, p-value = 0.006192
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -9.248813 -1.851187
## sample estimates:
## mean of x mean of y 
##     8.725    14.275
```
]

```r
with(
  gathmann.bt,
  t.test(aranei[gen == "ISO"],
    aranei[gen == "Bt"],
    alternative = "two.sided",
    var.equal = TRUE,
    conf.level = 0.95
  )
)
```

```
## 
## 	Two Sample t-test
## 
## data:  aranei[gen == "ISO"] and aranei[gen == "Bt"]
## t = -0.64498, df = 14, p-value = 0.5294
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.4595692  0.2470692
## sample estimates:
## mean of x mean of y 
##   0.46875   0.57500
```
]
]
]]
]

]

.footnote.f6[
Note significance test suggested is different in Achim Gathmann et al. (2006) “Impact of Bt Maize Pollen (MON810) on Lepidopteran Larvae Living on Accompanying Weeds.” Molecular Ecology 15: 2677–85.
]

---

# Confidence interval for two sample difference

]
.item[

* In the right, a 95% confidence interval for population mean difference is constructed repeatedly, assuming population mean difference is Normally distributed, from 100 samples of the same population.
* The population mean is zero. 
* Each confidence interval is calculated as 
`$$\bar{X} - \bar{Y} \pm \color{red}{t_{n-2, 0.975}}\times SE(\bar{X} - \bar{Y})$$`
where `$t_{n-2, 0.975}$` is `$t^*$` such that `$$P(t_{n-2} < t^*) =  0.975.$$`

]

---

# Two sample non-parametric tests

* Suppose that `$X$` and `$Y$` are randomly selected values from two populations.

`$$H_0: P(X > Y) = P(X < Y)$$`
.center[
vs.
]
`$$H_1: P(X>Y)\neq P(X<Y)$$`
* All observations are ranked.
* Test statistic is based on the sum of the ranks of one group.

]
.w-50[
.panelset[
.panel[.panel-name[thrip]
.f5[

```r
with(
  gathmann.bt,
  wilcox.test(thysan[gen == "ISO"],
    thysan[gen == "Bt"],
    alternative = "two.sided",
    conf.int = TRUE,
    conf.level = 0.95
  )
)
```

```
## 
## 	Wilcoxon rank sum exact test
## 
## data:  thysan[gen == "ISO"] and thysan[gen == "Bt"]
## W = 7, p-value = 0.006993
## alternative hypothesis: true location shift is not equal to 0
## 95 percent confidence interval:
##  -9.4 -2.4
## sample estimates:
## difference in location 
##                   -6.3
```
]
.panel[.panel-name[spider]
.f5[

```r
with(
  gathmann.bt,
  wilcox.test(aranei[gen == "ISO"],
    aranei[gen == "Bt"],
    alternative = "two.sided",
    conf.int = TRUE,
    conf.level = 0.95
  )
)
```

```
## 
## 	Wilcoxon rank sum test with continuity correction
## 
## data:  aranei[gen == "ISO"] and aranei[gen == "Bt"]
## W = 26.5, p-value = 0.5926
## alternative hypothesis: true location shift is not equal to 0
## 95 percent confidence interval:
##  -0.5999462  0.2000045
## sample estimates:
## difference in location 
##            -0.04998159
```
]]
]
]]
]

---

# Equivalence of tests to testing model parameters

```
##    gen thysan aranei
## 1   Bt   16.6   0.80
## 2   Bt   16.4   0.80
## 3   Bt   11.0   0.60
## 4   Bt   16.8   0.40
## 5   Bt   10.6   0.60
## 6   Bt   18.4   0.80
## 7   Bt   14.2   0.00
## 8   Bt   10.2   0.60
## 9  ISO    6.2   0.75
## 10 ISO   10.0   0.20
## 11 ISO   11.8   1.00
## 12 ISO   15.6   0.80
## 13 ISO    7.6   0.00
## 14 ISO    7.4   0.00
## 15 ISO    7.2   0.60
## 16 ISO    4.0   0.40
```

]]
.w-70.pl3[
`$$\texttt{thysan}_i = \beta_0 + \beta_1\mathbb{I}(\texttt{gen}_i=\texttt{ISO}) + e_i$$`
where `$e_i \sim NID(0, \sigma^2)$`.

* The least squares estimate for `$\widehat{\beta}_1 = \bar{X} - \bar{Y}.$`

```r
lm(thysan ~ gen, data = gathmann.bt) %>%
  confint("genISO", level = 0.95)
```

```
##            2.5 %    97.5 %
## genISO -9.248813 -1.851187
```
{{content}}
]
]

* Notice that the above confidence interval is the same confidence interval from the `$t$`-test!

---

# Revisiting .orange[Case study] .circle.bg-orange.white[4] Weight gain of calves .font_small[Part 1/3]

* 67 calves born in 1975 across 11 herds are fed of one of three diets with low, medium or high energy with their initial and final weights recorded.
.panelset[
.panel[.panel-name[📊]
<img src="images/week7B/diet-plot2-1.png" width="1008" style="display: block; margin: auto;" />

]
.panel[.panel-name[data]
.h100.scroll-sign.f4[

```r
data(urquhart.feedlot, package = "agridat")
df4 <- urquhart.feedlot %>%
  pivot_longer(c(weight1, weight2),
    names_to = "when",
    values_to = "weight"
  ) %>%
  mutate(
    when = factor(as.character(when),
      labels = c("initial", "final"),
      levels = c("weight1", "weight2")
    ),
    diet = factor(diet, levels = c("High", "Medium", "Low"))
  )
skimr::skim(df4)
```

```
## ── Data Summary ────────────────────────
##                            Values
## Name                       df4   
## Number of rows             134   
## Number of columns          5     
## _______________________          
## Column type frequency:           
##   factor                   2     
##   numeric                  3     
## ________________________         
## Group variables            None  
## 
## ── Variable type: factor ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate ordered n_unique top_counts               
## 1 diet                  0             1 FALSE          3 Low: 64, Med: 46, Hig: 24
## 2 when                  0             1 FALSE          2 ini: 67, fin: 67         
## 
## ── Variable type: numeric ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate  mean    sd  p0   p25  p50    p75 p100 hist 
## 1 animal                0             1  34    19.4   1  17.2  34    50.8   67 ▇▇▇▇▇
## 2 herd                  0             1  25.6  10.7   3  19    31    34     36 ▂▁▂▂▇
## 3 weight                0             1 863.  182.  530 705   842. 1027   1248 ▃▇▃▆▃
```
]]
.panel[.panel-name[R]

```r
ggplot(df4, aes(when, weight, color = diet, group = animal)) +
  geom_point(size = 3) +
  facet_wrap(~herd, nrow = 2) +
  geom_line() +
  labs(x = "", y = "Weight", color = "Diet")
```
]
]

---

# Revisiting .orange[Case study] .circle.bg-orange.white[4] Weight gain of calves .font_small[Part 2/3]

```r
coef(lm((weight2 - weight1) ~ diet, data = urquhart.feedlot))
```

```
## (Intercept)     dietLow  dietMedium 
##  332.666667   -4.666667  -33.971014
```
]
* The herd is thought to be an important factor contributing to the response. 
* Modelling the response as weight gain with diet and herd factor:
.f4[

```r
# herd needs to be factor not integer
dat4 <- mutate(urquhart.feedlot, herdf = factor(herd))
coef(lm((weight2 - weight1) ~ herdf + diet, data = dat4))
```

```
## (Intercept)      herdf9     herdf16     herdf19     herdf24     herdf31     herdf32     herdf33     herdf34     herdf35     herdf36     dietLow  dietMedium 
##  354.257353  -91.148529  -51.312039    7.410059  -63.221311   -4.666667  -51.189338  -36.083555  -15.179622   -3.423367  -34.974083    2.872233  -23.580408
```
]

]
.w-50.pl3[
* Last model is the same as modelling the final weight with the initial weight as a covariate with slope fixed to 1:
.f4[

```r
coef(lm(weight2 ~ offset(weight1) + herdf + diet,
  data = dat4
))
```

* Estimating slope for initial weight from the data:
.f4[

```r
coef(lm(weight2 ~ weight1 + herdf + diet,
  data = dat4
))
```

```
## (Intercept)     weight1      herdf9     herdf16     herdf19     herdf24     herdf31     herdf32     herdf33     herdf34     herdf35     herdf36     dietLow  dietMedium 
##  200.440174    1.243238  -79.102111  -51.238137   -6.864643  -75.406093  -33.044411  -56.517848  -62.152563  -23.610465  -29.219641  -75.555713    1.212633  -30.912720
```
]]
]

---

# Revisiting .orange[Case study] .circle.bg-orange.white[4] Weight gain of calves .font_small[Part 3/3]

]
.panel[.panel-name[R]

```r
dat4 <- lm(weight2 ~ weight1 + herdf + diet, data = dat4) %>% broom::augment()
ggplot(dat4, aes(.fitted, .resid)) +
  geom_point(data = select(dat4, -herdf), size = 2, color = "gray80") +
  geom_point(size = 2, aes(color = herdf)) +
  geom_hline(yintercept = 0) +
  labs(x = "Fitted values", y = "Residual") +
  scale_color_discrete_qualitative() +
  facet_wrap(~herdf, nrow = 2) +
  guides(color = FALSE) +
  theme(axis.text = element_blank())
```

]
]

---

# .orange[Case study] .circle.bg-orange.white[10] Soil nitrogen .font_small[Part 1/3]

.panelset[
.panel[.panel-name[📊]
<img src="images/week7B/soil-plot1-1.png" width="576" style="display: block; margin: auto;" />
]
.panel[.panel-name[data]
.h300.scroll-sign.f5[

```r
data(gomez.nitrogen, package = "agridat")
skimr::skim(gomez.nitrogen)
```

```
## ── Data Summary ────────────────────────
##                            Values        
## Name                       gomez.nitrogen
## Number of rows             96            
## Number of columns          4             
## _______________________                  
## Column type frequency:                   
##   factor                   3             
##   numeric                  1             
## ________________________                 
## Group variables            None          
## 
## ── Variable type: factor ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate ordered n_unique top_counts                    
## 1 trt                   0             1 FALSE          8 T1: 12, T2: 12, T3: 12, T4: 12
## 2 rep                   0             1 FALSE          4 R1: 24, R2: 24, R3: 24, R4: 24
## 3 stage                 0             1 FALSE          3 P1: 32, P2: 32, P3: 32        
## 
## ── Variable type: numeric ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
##   skim_variable n_missing complete_rate mean    sd   p0  p25  p50  p75 p100 hist 
## 1 nitro                 0             1 2.24 0.796 1.11 1.46 2.21 2.93 3.84 ▇▅▅▅▃
```

]
]
.panel[.panel-name[R]
.f5[

```r
ggplot(gomez.nitrogen, aes(trt, nitro, color = stage)) +
  geom_point(size = 3) +
  scale_color_discrete_qualitative() +
  labs(
    x = "Fertilizer treatment",
    y = "Soil nitrogen content (%)",
    color = "Growth stage"
  )
```
]]

]

]
.w-50[
* Soil nitrogen content with 8 different fertilizer treatment is measured at 3 growth stage:
  * P1 = 15 days post transplanting
  * P2 = 40 days post transplanting
  * P3 = panicle initiation
{{content}}

]
]

--
* Clearly the growth stage affects the soil nitrogen content but this makes it hard to compare the fertilizer treatments.
{{content}}
--
* Let's model the nitrogen content as:

```r
lm(nitro ~ stage + trt, 
  data = gomez.nitrogen)
```

---

# .orange[Case study] .circle.bg-orange.white[10] Soil nitrogen .font_small[Part 2/3]

```r
fit1 <- lm(nitro ~ stage, data = gomez.nitrogen)
fit1data <- broom::augment(fit1) %>%
  left_join(gomez.nitrogen, by = c("nitro", "stage")) %>%
  mutate(trt = fct_reorder(trt, .resid))
ggplot(fit1data, aes(trt, .resid)) +
  geom_boxplot() +
  labs(
    x = "Fertilizer treatment",
    y = "Residual of fit1"
  )
```

]]
.w-50.pl3[
.f5[
{{content}}

]]
]

* Here we expect no pattern:

```r
fit2 <- lm(nitro ~ stage + trt,
  data = gomez.nitrogen
)
fit2data <- broom::augment(fit2) %>%
  mutate(trt = fct_reorder(trt, .resid))
ggplot(fit2data, aes(trt, .resid)) +
  geom_boxplot() +
  labs(
    x = "Fertilizer treatment",
    y = "Residual of fit2"
  )
```

---

# .orange[Case study] .circle.bg-orange.white[10] Soil nitrogen .font_small[Part 3/3]
.flex[
.w-50[
.h300.scroll-sign.f5[

```r
library(emmeans)
confint(pairs(emmeans(fit2, "trt"), adjust = "none"))
```

```
##  contrast estimate    SE df lower.CL  upper.CL
##  T1 - T2   -0.2117 0.116 86  -0.4420  0.018654
##  T1 - T3   -0.3375 0.116 86  -0.5678 -0.107180
##  T1 - T4   -0.2308 0.116 86  -0.4612 -0.000513
##  T1 - T5   -0.0717 0.116 86  -0.3020  0.158654
##  T1 - T6   -0.1492 0.116 86  -0.3795  0.081154
##  T1 - T7   -0.3592 0.116 86  -0.5895 -0.128846
##  T1 - T8   -0.2333 0.116 86  -0.4637 -0.003013
##  T2 - T3   -0.1258 0.116 86  -0.3562  0.104487
##  T2 - T4   -0.0192 0.116 86  -0.2495  0.211154
##  T2 - T5    0.1400 0.116 86  -0.0903  0.370320
##  T2 - T6    0.0625 0.116 86  -0.1678  0.292820
##  T2 - T7   -0.1475 0.116 86  -0.3778  0.082820
##  T2 - T8   -0.0217 0.116 86  -0.2520  0.208654
##  T3 - T4    0.1067 0.116 86  -0.1237  0.336987
##  T3 - T5    0.2658 0.116 86   0.0355  0.496154
##  T3 - T6    0.1883 0.116 86  -0.0420  0.418654
##  T3 - T7   -0.0217 0.116 86  -0.2520  0.208654
##  T3 - T8    0.1042 0.116 86  -0.1262  0.334487
##  T4 - T5    0.1592 0.116 86  -0.0712  0.389487
##  T4 - T6    0.0817 0.116 86  -0.1487  0.311987
##  T4 - T7   -0.1283 0.116 86  -0.3587  0.101987
##  T4 - T8   -0.0025 0.116 86  -0.2328  0.227820
##  T5 - T6   -0.0775 0.116 86  -0.3078  0.152820
##  T5 - T7   -0.2875 0.116 86  -0.5178 -0.057180
##  T5 - T8   -0.1617 0.116 86  -0.3920  0.068654
##  T6 - T7   -0.2100 0.116 86  -0.4403  0.020320
##  T6 - T8   -0.0842 0.116 86  -0.3145  0.146154
##  T7 - T8    0.1258 0.116 86  -0.1045  0.356154
## 
## Results are averaged over the levels of: stage 
## Confidence level used: 0.95
```
]
]
.w-50.f4[
<img src="images/week7B/unnamed-chunk-7-1.png" width="576" style="display: block; margin: auto;" />
* From above, the 6 pairs of treatments: T3 & T5, T1 & T4, T1 & T8, T6 & T7, T1 & T3, T1 & T7 are significantly different. 
* These confidence intervals are constructed *without taking any regard for others*.

]

]
---

# Controlling the family-wise error rate

**Unadjusted**

* Each interval has been constructed using a procedure so that when the model is correct,
the probability that the "correct" population contrast is covered is 0.95. . . individually.

`$$\bar{X} - \bar{Y} \pm \color{red}{t_{n-t,1 - \alpha/2}}\times SE(\bar{X} - \bar{Y})$$`
where `$\alpha = 0.05$` and `$t$` is the number of treatments.

* But, what is the probability that all intervals cover their corresponding true values simultaneously?

]
.w-50.pl3[
**Bonferroni adjustment**

* We can adjust the individual `$100(1-\alpha)\%$` confidence intervals so

`$$\bar{X} - \bar{Y} \pm \color{red}{t_{n-t,1 - \alpha/(2m)}}\times SE(\bar{X} - \bar{Y})$$`
where `$m$` is the number of pairwise comparisons.

* So for 8 treatments, the number of pairwise comparisons is

```r
choose(8, 2)
```

```
## [1] 28
```

]
]

---

# Bonferroni adjusted confidence interval
.flex[
.w-50[
.h300.scroll-sign.f5[

```r
confint(pairs(emmeans(fit2, "trt"),
  adjust = "bonferroni"
))
```

```
##  contrast estimate    SE df lower.CL upper.CL
##  T1 - T2   -0.2117 0.116 86   -0.585   0.1619
##  T1 - T3   -0.3375 0.116 86   -0.711   0.0361
##  T1 - T4   -0.2308 0.116 86   -0.604   0.1427
##  T1 - T5   -0.0717 0.116 86   -0.445   0.3019
##  T1 - T6   -0.1492 0.116 86   -0.523   0.2244
##  T1 - T7   -0.3592 0.116 86   -0.733   0.0144
##  T1 - T8   -0.2333 0.116 86   -0.607   0.1402
##  T2 - T3   -0.1258 0.116 86   -0.499   0.2477
##  T2 - T4   -0.0192 0.116 86   -0.393   0.3544
##  T2 - T5    0.1400 0.116 86   -0.234   0.5136
##  T2 - T6    0.0625 0.116 86   -0.311   0.4361
##  T2 - T7   -0.1475 0.116 86   -0.521   0.2261
##  T2 - T8   -0.0217 0.116 86   -0.395   0.3519
##  T3 - T4    0.1067 0.116 86   -0.267   0.4802
##  T3 - T5    0.2658 0.116 86   -0.108   0.6394
##  T3 - T6    0.1883 0.116 86   -0.185   0.5619
##  T3 - T7   -0.0217 0.116 86   -0.395   0.3519
##  T3 - T8    0.1042 0.116 86   -0.269   0.4777
##  T4 - T5    0.1592 0.116 86   -0.214   0.5327
##  T4 - T6    0.0817 0.116 86   -0.292   0.4552
##  T4 - T7   -0.1283 0.116 86   -0.502   0.2452
##  T4 - T8   -0.0025 0.116 86   -0.376   0.3711
##  T5 - T6   -0.0775 0.116 86   -0.451   0.2961
##  T5 - T7   -0.2875 0.116 86   -0.661   0.0861
##  T5 - T8   -0.1617 0.116 86   -0.535   0.2119
##  T6 - T7   -0.2100 0.116 86   -0.584   0.1636
##  T6 - T8   -0.0842 0.116 86   -0.458   0.2894
##  T7 - T8    0.1258 0.116 86   -0.248   0.4994
## 
## Results are averaged over the levels of: stage 
## Confidence level used: 0.95 
## Conf-level adjustment: bonferroni method for 28 estimates
```
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<img src="images/week7B/unnamed-chunk-10-1.png" width="576" style="display: block; margin: auto;" />
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* Now none are significantly different.
* Note: Bonferroni adjustment is quite conservative.

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<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a>.

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Lecturer: *Di Cook*

<i class="fas fa-envelope"></i>  ETC5521.Clayton-x@monash.edu

<i class="fas fa-calendar-alt"></i> Week 7 - Session 2

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